MPC – 1
PERMUTATION AND COMBINATION
Permutations are arrangements and combinations are selections.
C1
Fundamental Principle and Counting :
(i)
Principle of Multiplication :
If an event can occur in ‘m’ different ways, following which another event can occur in ‘n’ different ways,
then total number of different ways of simultaneous occurrence of both the events in a definite order is
m × n.
(ii)
Principle of Addition :
If an event can occur in ‘m’ different ways, and another event can occur in ‘n’ different ways, then exactly
one of the events can happen in m + n ways.
Practice Problems :
1.
How many numbers are there between 100 and 1000 in which all the digits are distinct ?
2.
How many 3-digit numbers can be formed without using the digits 0, 2, 3, 4, 5 and 6 ?
3.
How many odd numbers less than 1000 can be formed by using the digits 0, 2, 5, 7 when the
repetition of digits is allowed ?
4.
How many numbers are there between 100 and 1000 such that 7 is in the unit’s place ?
5.
How many numbers are there between 100 and 1000 such that at least one of their digits is 7 ?
6.
For a set of five true or false questions, no student has written the all correct answer and no two
students have given the same sequence of answers. What is the maximum number of students in the
class for this to be possible ?
[Answers : (1) 648 (2) 64 (3) 32 (4) 90 (5) 90, 81, 252 (6) 31]
C2
Arrangement :
If nPr or P(n, r) denotes the number of permutations of n different things, taking r at a time, then
n
Pr  n(n  1)(n  2)....(n  r  1) 
n! where 1  r  n.
(n  r )!
Important Theorems :
1.
The number of all permutations of n different things taken all at a time is given by nPn = n !.
2.
The number of all permutations of n different things taken r at a time, when a particular thing is
to be always included in each arrangement is r . n – 1Pr – 1.
3.
The number of permutations of n different things taken r at a time, when a particular thing is
never taken in each arrangement, is n – 1Pr.
4.
n
5.
n
6.
The number of permutations of ‘n’ things, taken all at a time, when ‘p’ of them are similar and of
one type, q of them are similar and of another type, ‘r’ of them are similar and of a third type and
Pr = n – 1Pr + r . n – 1Pr – 1.
Pr = n . n – 1Pr – 1.
the remaining n – (p + q + r) are all different is
7.
n!
.
p! q! r!
The number of permutations of n different objects, taken r at a time when each may be repeated
any number of times in each arrangement, is nr.
Important Points :
Einstein Classes,
(i)
factorials of negative integers are not defined.
(ii)
0!=1!=1;
(iii)
n
(iv)
(2n) ! = 2n . [1 . 3 . 5 . 7 .... (2n – 1)] n !
Pn = n ! = n . (n – 1) !
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 2
Practice Problems :
1.
2.
In how many ways can the letters of the word PERMUTATIONS be arranged if the
(i)
words start with P and end with S,
(ii)
(iii)
there are always 4 letters between P and S ?
vowels are all together,
Prove that 33 ! is divisible by 215.
What is the largest integer n such that 33 ! is divisible by 2n ?
3.
How many different signals can be given with 5 different flags by hoisting any number of them at a
time ?
4.
It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How
many such arrangements are possible ?
5.
How many words can be formed from the letters of the word ‘DAUGHTER’ so that
(i)
the vowels always come together;
(ii)
the vowels are never together ?
6.
Find the number of ways in which the letters of the word ‘MACHINE’ can be arranged such that the
vowels may occupy only odd positions.
7.
(i)
Find how many arrangements can be made with the letters of the word
‘MATHEMATICS’.
(ii)
In how many of them are the vowels together ?
8.
In how many ways can 6 rings of different type be had in 4 fingers ?
[Answers : (1) (i) 1814400 (ii) 2419200 (iii) 25401600 (3) 325 (4) 2880 (5) (i) 4320 (ii) 36000 (6) 576
(7) (i) 4989600 (ii) 120960 (8) 4096]
C3
Circular Permutation :
The number of circular permutations of n different things taken all at a time is (n – 1) !
If clockwise and anti-clockwise circular permutations are considered to be same, then it is
(n  1)! .
2
Number of circular permutation of n things when p alike and the rest different taken all at a time distinguishing clockwise and anticlockwise arrangement is
(n  1)!
p!
Practice Problems :
1.
In how many ways can 8 students be arranged in (i) a line, (ii) a circle ?
2.
If 20 persons were invited for a party, in how many ways can they and the host be seated at a circular
table ?
In how many of these ways will two particular persons be seated on either side of the host ?
3.
In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two
women are adjacent ?
4.
A round table conference is to be held delegates of 20 countries. In how many ways can they be
seated if two particular delegates may wish to sit together ?
5.
In how many ways can 7 persons sit around a table so that all shall not have the same neighbour in
any two arrangements ?
6.
3 boys and 3 girls are to be seated around a table in a circle. Among them, the boy X does not want
any girl neighbour and the girl Y does not want any boy neighbour. How many such arrangements
are possible ?
7.
Find the number of ways in which 8 different beads can be arranged to form a necklace.
[Answers : (1) (i) 40320 (ii) 5040 (2) (2 × 18 !) (3) 144 (4) (2 × (18 !) (5) 360 (6) 4 (7) 2520]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 3
C4
Selection :
n
If nCr [C(n, r) or   denotes the number of combinations of n different things taken r at a time, then
r
n
Cr 
n
n!
P
 r where r  n
r! (n  r )! r!
Important Theorems :
(i)
n
Cr = nCn – r
(ii)
n
(iii)
n
Cr + nCr – 1 = n + 1Cr
Cr = 0 if r
 {0, 1, 2, 3,....., n}
Practice Problems :
1.
How many chords can be drawn through 21 points on a circle ?
2.
In how many ways can a cricket team be chosen out of a batch of 15 players, if
3.
(a)
there is no restriction on the selection;
(b)
a particular player is always chosen;
(c)
a particular player is never chosen ?
A committee of 5 is to be formed out of 6 men and 4 ladies. In how many ways can this be done, when
(a)
at least 2 ladies are included;
(b)
at most 2 ladies are included ?
4.
An examination paper containing 12 questions consists of two parts, A and B. Part A contains
7 questions and part B contains 5 questions. A candidate is required to attempt 8 questions, selecting
at least 3 from each part. In how many ways can the candidate select the questions ?
5.
How many diagonals are there in a polygon of n sides ?
6.
There are 10 points in a plane, no three of which are in the same straight line. except 4 points, which
are collinear.
Find :
(i) the number of lines obtained from the pairs of these points;
(ii) the number of triangles that can be formed with vertices as these points.
7.
In an examination, a candidate has to pass in each of the 5 subjects. In how many ways can he fail ?
8.
In how many ways can 21 books on English and 19 books on Hindi be placed in a row on a shelf so
that two books on Hindi may not be together ?
[Answers : (1) 210 (2) (a) 1365 (b) 1001 (c) 364 (3) (a) 186 (b) 180 (4) 420 (5)
1
n(n – 3) (6) (i) 40
2
(ii) 116 (7) 31 (8) 1540 ]
C5
Formation of Groups :
Number of ways in which (m + n + p) different things can be divided into three different groups containing
m, n and p things respectively is
(m  n  p )!
,
m! n! p!
If m = n = p and the groups have identical qualitative characteristic then the number of groups =
( 3n)!
.
n! n! n!3!
However, if 3n things are to be divided equally among three people then the number of ways =
Einstein Classes,
( 3n )!
(n! ) 3
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 4
C6
C7
C8
Selections of one or more objects
(a)
Number of ways in which atleast one object be selected out of ‘n’ distinct objects is
n
C1 + nC2 + nC3 + .... + nCn = 2n – 1
(b)
Number of ways in which atleast one object may be selected out of ‘p’ alike objects of one type
‘q’ alike objects of second type and ‘r’ alike of third type is (p + 1) (q + 1) (r + 1) – 1
(c)
Number of ways in which atleast one object may be selected from ‘n’ objects where ‘p’ alike of
one type ‘q’ alike of second type and ‘r’ alike of third type and rest n – (p + q + r) are different,
is (p + 1) (q + 1) (r + 1) 2n – (p + q + r) – 1
Multinominal Theorem :
Number of ways in which it is possible to make a selection from m + n + p = N things, where p are alike of
one kind, m alike of second kind & n alike of third kind taken r at a time is given by coefficient of xr in the
expansion of (1 + x + x2 + .... + xp) (1 + x + x2 + .... + xm) (1 + x + x2 + .... + xn).
Important Result :
(i)
Method of fictious partition : Number of ways in which n identical things may be distributed
among p persons if each person may receive none, one or more things is, n + p – 1Cn.
(ii)
Let N = pa . qb . rc .... where p, q, r .... are distinct primes a, b, c .... are natural numbers then :
(a)
The total numbers of divisors of N including 1 & N is = (a + 1) (b + 1) (c + 1)...
(b)
The sum of these divisors is =
(p0 + p1 + p2 + .... + pa) (q0 + q1 + q2 + .... + qb) (r0 + r1 + r2 + .... + r0)....
(c)
Number of ways in which N can be resolved as a product of two factors is
=
(d)
(iii)
(iv)
1
(a  1)(b  1)(c  1).... if N is not a perfect square
2
1
[(a  1)(b  1)(c  1)....  1] if N is a perfect square
2
Number of ways in which a composite number N can be resolved into two factors which are
relatively prime (or coprime) to each other is equal to 2n – 1 where n is the number of different
prime factors in N.
Let there be ‘n’ types of objects, with each type containing atleast r objects. Then the number of
ways of arranging r objects in a row is nr.
Dearrangement : Number of ways in which ‘n’ letters can be put in ‘n’ corresponding
envelopes such that no letter goes to correct envelope is
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1 1 1 1
1

n!  1     ....  ( 1)n 
n! 
 1! 2! 3! 4!
Miscellaneous Problems :
(i)
A gentleman has 6 friends to invite. Inhow many ways can he send invitation cards to
them, if he has three servants to carry the cards ?
(ii)
A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the
position of rest. What is the total number of signals that can be made ?
Find the sum of all the four-digit numbers that can be formed with the digits 3, 2, 3, 4.
There are 10 points in a plane of which only 4 are collinear. How many different triangles can be
formed with these points as vertices ?
In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least
one woman has to be necessarily selected ?
In how many ways can 12 different things be equally distributed among 4 persons ? If they are
divided into 4 groups instead of giving away to 4 persons, what will be the number of ways ?
A boat has a crew of 10 men of which 3 can row only on one side and 2 only on the other. In how
many ways can the crew be arranged ?
How many integers between 1 and 1000000 have the sum of the digits equal to 18.
How many three digit numbers are of the form xyz with x < y, z < y and x  0.
How many words, with or without meaning, can be formed using all the letters of the word
EQUATION at a time so that the vowels and consonents occur together ?
It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How
many such arrangements are possible ?
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are
together ?
[Answers : (1) (i) 729 (ii) 1023 ways (2) 3552 (3) 116 (4) 246 (5) 15400 (6) 144000 (7) 25927 (8) 240
(9) 1440 (10) 2880 (11) 151200]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 5
SINGLE CORRECT CHOICE TYPE
1.
2.
3.
4.
5.
6.
If n + 2C8 : n – 2 P4 = 57 : 16. The value of n is
(a)
17
(b)
18
(c)
19
(d)
20
A man has 7 relatives, 4 of them are ladies and 3
gentlemen; his wife has also 7 relatives, 3 of them
are ladies and 4 gentlemen. The number of ways
can they invite a dinner party of 3 ladies and 3
gentlemen so that there are 3 of them man’s
relatives and 3 of the wife’s relatives is
(a)
485
(b)
495
(c)
525
(d)
550
7.
8.
The number diagonals are there in a polygon with
20 sides is
(a)
170
(b)
180
(c)
340
(d)
360
The letters of the word SURITI are written in all
possible orders and these words are written out as
in a dictionary. The rank of the word SURITI is
9.
(a)
20
(b)
25
(c)
30
(d)
35
There are four balls of different colours and four
boxes of colours, same as those of the balls. The
number of ways in which the balls, one each in a
box, could be placed such that a ball does not go to
a box of its own colour is
(a)
9
(b)
8
(c)
7
(d)
6
Number of zeros at the end of 300! is equal to
(a)
75
(b)
89
(c)
74
(d)
98
(a)
225
(b)
231
The total number of six digit numbers x1 x2 x3 x4 x5
x6 having the property that x1 < x2  x3 < x4 < x5  x6
is equal to
(c)
236
(d)
242
(a)
10
(c)
11
In a certain test, ai students gave wrong answers to
at least i question where i = 1, 2, .... k. No student
gave more than k wrong answers. The total
number of wrong answer given is
(a)
2n
(b)
2n – 1
(c)
2n + 1
(d)
none
10.
Total number of ways in which six ‘+’ and four ‘–’
signs can be arranged in a line such that no two
‘–’ signs occur together is
11.
Let A be a set of n distinct elements. Then the total
number of distinct onto functions from A to A is
12.
n
(a)

n
C r (r ) n
r 1
n
(b)

n
Cr
C6
(b)
12
C6
(d)
none of these
C6
Total number of ways of selecting two numbers
from the set {1, 2, 3, 4,....., 3n} so that their sum is
divisible by 3 is equal to
(a)
2n 2  n
2
(b)
3n 2  n
2
(c)
2n2 – n
(d)
3n2 – n
Total number of numbers that are less than 3 × 108
and can be formed using the digits 1, 2, 3, is equal
to
(a)
1 9
( 3  4 .3 8 )
2
(b)
1 9
( 3  3)
2
(c)
1
( 7.3 8  3 )
2
(d)
1 9
(3  3  38 )
2
r 1
n
(c)
 (1)
n r n
r 1
(d)
none
Einstein Classes,
C r (r ) n
13.
There are two sets A = {a 1 , a 2, a 3,....,a m } and
B = {b1, b2, b3,...,bn} of real numbers. The minimum
number of ordered pairs (x, y), x  A, y  B that
must be written, is so that two pairs will be identical
(a)
mn + 1
(b)
mn
(c)
mn – 1
(d)
none
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 6
14.
The number of ways in which a set A where
n(A) = 12 can be partitioned in three subsets
P, Q, R each of 4 elements if P  Q  R = A,
P  Q = , Q R = , R  P = , is
12!
(a)
( 4! )
20.
n
n
n
Cr + 1 + Cr – 1 + 2 × Cr is equal to
(a)
n+2
(b)
n+1
(c)
n+1
(d)
n+2
Cr + 1
Cr + 1
12!
(b)
3
( 3! ) 4
n
21.
The value of

4n 1
Cr
Cr

C j  4n 1 C 2n  j is
j 0
12!
(c)
(4! )
3
.
1
3!
n
15.
In the identity
(d)
Ak
none of these
n!
 x  k  x(x  1)(x  2)....(x  n)
22.
k 0
then value of Ak is
16.
17.
18.
(a)
n
(b)
n
Ck
Ck + 1
k
(a)
24n + 4n + 1Cn
(b)
24n + 1
(c)
24n + 1 + 4n + 1Cn
(d)
24n
The number of divisors of the form 2n – 1(n  2) of
the number 2 p 3 q 4 r 5 s, where p, q, r, s belong to N,
is
(a)
qs + q + s + 1
(b)
(p + 1) (q + 1) (r + 1) (s + 1) – 1
(c)
qs + q + s
(d)
qs
n
(c)
(–1) · Ck
(d)
(–1)k – 1 · nCk – 1
23.
Let C1, C2, .... Cn, ... be a sequence of concentric
circles. The nth circle has the radius n and it has n
openings. A point P starts travelling on the smallest
circle C1 and leaves it at an opening along the
normal at the point of opening to reach the next
circle C2. Then it moves on the second circle C2 and
leaves it likewise to reach the third circle C3 and so
on. The total number of different paths in which
the point can come out of the nth circle is
(a)
2n · n !
(b)
2n – 1 · n !
(c)
n!
(d)
2n – 1 · (n – 1) !
24.
A flag is to be coloured in four stripes by using 6
different colours, no two consecutive stripes being
of the same colour. This can be done in
(a)
1500 ways
(b)
750 ways
(c)
64 ways
(d)
none of these
In a college of 300 students, every student reads 5
newspaper and every newspaper is read by
60 students. The number of newspaper is
(a)
at least 30
(b)
at most 20
(c)
exactly 25
(d)
none of these
25.
A word has 4 identical letters and some different
letters. If the total number of words that can be
made with the letters of the word be 210 then the
number of different letters in the word is
(a)
3
(b)
5
(c)
4
(d)
7
The number of ways in which 10 gentleman sit
round a table so that in no two arrangements they
have the same neighbours is
(a)
1
9!
2
(b)
1
8!
2
(c)
1
7!
2
(d)
none
The number of seven letter words can be formed
by using the letters of the word SUCCESS so that
no two C and no two S are together is
(a)
76
(b)
86
(c)
96
(d)
106
n
19.
The value of
(
n 1
C j  n C j ) is equal to
j 1
(a)
2n
(b)
2n + 1
(c)
3 · 2n
(d)
2n – 1
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 7
EXCERCISE BASED ON NEW PATTERN
6.
COMPREHENSION TYPE
Comprehension-1
A triangle is called an integer triangle if all the sides
are integers. Let x, y, z are sides of an integer
triangle and we can assume x  y  z (any other
permutation will yield same triangle). Since sum of
two sides is greater than the third side therefore if z
is fixed x + y will vary from z + 1 to 2z. The number
of such integer triangles can be found by finding
integer solutions of
The number of integer isosceles or equilateral
triangles none of whose sides exceed 2c must be
(a)
(c)
2.
3.
z2
3z2
(b)
(d)
2z2
(a)
2z  1
2
(b)
2z  1
2
(c)
3z  1
2
(d)
3z  1
2
3z  5
2
(b)
(A)
(B)
(c)
(d)
(1 – x)–n = n –1C0 + nC1x + n + 1C2x2 + .... + n + rCr + 1xr + 1
+ .... to , where |x| < 1.
The number of positive integral solution of the
equation 1 + 2 + 3 + 4 = 10 is
(a)
9
C6
(b)
10
C4
(c)
9
C4
(d)
10
C5
The number of negative integral solutions of the
equation 1 + 2 + 3 + 4 + 5 + 10 = 0 is
(a)
10
(c)
9
C4
C5
Einstein Classes,
C4
C2
Column - A
Column - B
the word neither begin
(p)
43200
(q)
151200
(r)
12 !
6 !2 !
(s)
83  10!
24
the vowels and
(C)
the vowels are always
consecutive
(D)
the order of vowels
does not change
Matching-2
Five balls are to be placed in three boxes. Each can
hold all the five balls. In how many different ways
can we place the balls so that no box remains empty,
if
Column-A
Column-B
balls and boxes are all
(p)
6
(q)
150
(r)
2
(s)
25
different
(B)
balls are identical but
boxes are different
To find the number of positive integral solution of
the equation  +  +  + ..... + m = n, where n is
the natural number, we have to find the coefficient
of xn in the expansion of (x + x2 + x3 + .... to )m
when |x| < 1. Also we have the expansion
5.
(d)
9
the words
Comprehension-2
4.
10
consonants alternate in
3z  2
2
3z  2
2
C3
(b)
with I nor end with E
(A)
z
2
(c)
9
C3
How may words can be made with letters of the
word INTERMEDIATE if
If z is even, the number of integer isosceles or
equilateral triangle whose sides are x, y, z, x  y  z
must be
(a)
8
Matching-1
3z 2
2
If z is fixed and odd, the number of integer
isosceles or equilateral triangle whose sides are
x, y, z, x  y  z must be
(a)
MATRIX-MATCH TYPE
x + y = z + 1, x + y = z + 2,......,x + y = 2z.
1.
The number of ways in which 10 identical pens can
be distributed among 3 students so that each gets
at least one pen is
(b)
10
(d)
9
C5
(C)
balls are different but
boxes are identical
(D)
balls as well as boxes
are identical
MULTIPLE CORRECT CHOICE TYPE
1.
Choose the correct statement from the following :
(a)
n
(b)
n
Pr = (n – r + 1) × nPr – 1
Pr = n – 1Pr + r. n – 1Pr – 1
n
(c)

k
C r  ( n 1 C r  1  m C r  1 )
k m
(d)
none
C6
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 8
2.
3.
4.
5.
Different car licence plates can be constructed
using three letters of the English alphabet followed
by a three digit number. Then
(a)
The total number of licence plates is
(26)3999 if repetitions are allowed
(b)
The total number of licence plates is
26
P3 10P3 if repetitions are not allowed
(c)
The total number of licence plates is
(26)3999 if repetitions are not allowed
(d)
The total number of licence plates is
26
P3 10P3 if repetitions are allowed
There are n points in a plane out of these points no
three are in the same straight line except p points
which are collinear. Then
(a)
The number of straight lines can be
formed by joining them is nC2 – pC2 + 1
(b)
The number triangles can be formed by
joining them is nC3 – pC3.
(c)
The number of straight lines can be
formed by joining them is nC2 – pC2
(d)
none of these
Consider the following inequality :
x–1
C4 – x – 1C3 –
5 x–2
( P2) < 0, x  N
4
The values of ‘x’ may be
8.
(a)
5
(c)
7
If
n+1
(a)
n
Cr + 1 : Cr :
6
(d)
8
Cr – 1 = 11 : 6 : 3, then
n = 10
(b)
n = 12
(c)
9.
10.
r=5
(d)
r=6

   
Let a  i  j  k and let r be a variable vector such
  

that r · i , r · j and r ·k are positive integers. If

 
r · a  12 then the number of values of r is
(a)
12
(c)
12
C9 – 1
C9
(b)
12
(d)
none of these
C3
The product fo r consecutive integers is divisible
by
r 1
(a)
(a)
The total number of such digits formed
is 840
Assertion-Reason Type
(b)
The total number of even digits formed
is 360
(c)
The number of such digits exactly
divisible by 4 is 200
(d)
The number of such digits are exactly
divisible by 25 is 40
(c)
the number of arrangements such that
no two ladies sit side by side is 2880
(c)
the number of arrangements such that
there is no restriction is 387520
(d)
the number of arrangements such that
no two ladies sit side by side is 2570
There are different selections of 4 books can be
made from 10 different books. Then
(a)
the number of different selections if there
is no restricton is 210
(b)
the number of different selections if two
particular books are always selected is 28
(c)
the number of different selections if two
particular books are never selected is 70
(d)
none
(b)
k
r!
(d)
none of these
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
the number of arrangements such that
there is no restriction is 362880
(b)
r
k 1
There are 5 gentlemen and 5 ladies sit around a
table. Then
Einstein Classes,
(b)
n–1
A number of four different digits is formed with
the help of the digits 1, 2, 3, 4, 5, 6, 7 in all possible
ways. Then
(a)
6.
7.
1.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
(B)
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-1 :
n–1
C3 + n – 1C4 > nC3 if n > 7.
STATEMENT-2 : n + 1Cm + 1 = nCm + nCm + 1.
2.
STATEMENT-1 : Two positive inegers n, r
cannot be found such that nCr, nCr + 1, nCr + 2 are in
G.P.
STATEMENT-2 : Two positive inegers n, r
cannot be found such that nCr, nCr + 1, nCr + 2 are in
A.P.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 9
3.
STATEMENT-1 : The number of ways to select 3
numbers in AP from the first n natural numbers
is
1
(n – 1)2.
4
STATEMENT-2 : The number of ways to select
three numbers in AP from the first 2n natural
numbers is n(n – 1).
4.
STATEMENT-1 : The product of r consecutive
positive even integers is divisible by 2r × r !.
STATEMENT-2 : The product of any r
consecutive natural numbers is always divisible
by r !.
STATEMENT-1 : (n !) ! is divisible by (n !)(n – 1) !.
5.
STATEMENT-2 : 1 ! + 2 ! + 3 ! + .... + n ! cannot
be a perfect square for any n  N, n  4.
(Answers) EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
1.
c
2.
d
3.
b
4.
a
2.
[A-q; B-p; C-s; D-r]
5.
c
6.
d
5.
a, b
6.
MATRIX MATCH TYPE
1.
[A-s; B-p; C-q; D-r]
MULTIPLE CORRECT CHOICE TYPE
1.
a, b, c
2.
a, b
3.
a, b
4.
a, b, c, d
7.
a, b, c, d 8.
a, c
9.
b, c
10.
a, b, c
3.
D
4.
B
a, b, c
ASSERTION-REASON TYPE
1.
A
2.
Einstein Classes,
B
5.
B
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 10
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
India and South Africa play one day international
series until one team wins 4 matches. No match ends
in a draw. Find in how many ways the series can be
won.
12.
n different objects are arranged in a row. In how
many ways can 3 objects be selected so that
(i)
all the three objects are consecutive
(ii)
all the three objects are not consecutive
2.
How many different numbers which are smaller
than 2 × 108 and are divisible by 3, can be written
by means of the digits 0, 1 and 2 ?
3.
The number of ways in which 25 identical things
can be distributed among five persons when each
gets odd number of things is
4.
John has x children by his first wife. Mary has
(x + 1) children by her first husband. They marry
and have children of their own. The whole family
has 24 children. Assuming that two children of the
same parent do not fight. Prove that the maximum
number of fights that can be take place is 191.
5.
Show that the number of ways in which 2n things
of one sort, 2n of another sort and 2n of a third
sort can be divide between two persons, giving 3n
things to each, is 3n2 + 3n + 1.
15.
Show that the number of different selections of 5
letters from five As, four Bs, three Cs, two Ds and
one E is 71.
Find the number of non-negative integral solutions
to the system of equations x + y + z + u + t = 20 and
x + y + z = 5.
16.
An eight-oared boat is to be manned by a crew
chosen from 11 men of whom 3 can steer but
cannot row and the rest cannot steer. In how many
ways can be crew be arranged if two of the men
can only row on bow side ?
17.
A train going from Calcutta to Delhi stops at 7
intermediate stations. Five persons enter the train
during the journey with five different tickets of the
same class. How many different set of tickets they
could have had ?
18.
There are two bags each containing m balls. Find
the number of ways in which equal number of balls
can be selected from both bags if at least one ball
from each bag is to be selected.
19.
A gentleman invites a party of 10 friends to a
dinner and there are 6 places at one round table
and the remaining 4 at another. Prove that the
number of ways in which he can arrange them
among themselves is 151200.
20.
Show that the total number of permutations of n
different things taken not more than r at a time,
when each thing may be repeated then any
6.
7.
In how many ways can a committee of 5 women
and 6 men by chosen from 10 women and 8 mer if
Mr. A refuses to serve on the committee if Ms. B is
a member.
8.
Prove (without using the binomial theorem) that
13.
From 6 gentleman and 4 ladies, a committee of 5 is
to be formed. In how many ways can this be done if
the committee is to include at least one lady and if
two particular ladies refuse to serve on the same
committee ?
14.
Six “X’s” have to be placed in the square of the
figure given below, such that each row contains at
least one X.
In how many different ways can this be done ?
n
(i)
2 n
k . C
k
 2 n  2 n (n  1)
k 1
n
(ii)
3 n
k . C
k
2
n 3
2
n (n  3)
k 1
9.
A man invites a party of (m + n) friends to dinner
and places m at one round and n at another. Find
the number of ways of arranging the guests.
10.
The streets of a city are arranged like the lines of a
chess-board. There are m streets running North and
South and n East and West. Find the number of
ways in which a man can travel from the N.W. to
the S.E. corner, going the shortest possible distance.
11.
There is a polygon of n sides (n > 5). Triangles are
formed by joining the vertices of the polygon. How
many triangles are there ? Also prove that the
number of these triangles which have no side
common with any of the sides of the polygon is
number of times is
n (n r  1)
.
(n  1)
1
n(n  4)(n  5).
6
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 11
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
There are p intermediate stations on a railway line
from one terminus to another. In how many ways a
train can stop at 3 of these intermediate stations if
no of two of these stopping stations are to be
consecutive.
2.
There are 2n guests at a dinner party. Supposing
that the master and mistress of the house have fixed
seats opposite one another, and that there are two
specified guests who must not be placed next to one
another, find the number of ways in which the
guests can be placed.
3.
4.
A is a set containing n elements. A subset P of A is
chosen. The set A is reconstructed by replacing the
element of P. A subset Q of A is again chosen. Find
the number of ways of choosing P and Q so that
P  Q contains exactly two elements.
5.
6 balls marked as 1, 2, 3, 4, 5 and 6 are kept in a
box. Two players A and B start to take out 1 ball at
a time from the box one after another without
replacing the ball till the game is over. The number marked on the ball is added each time to the
previous sum to get the sum of numbers marked
on the balls taken out. If this sum is even then 1
point is given to the player. The first player to get 2
point is declared winner. At the start of the game
the sum is 0.1. If A starts to take out the ball, then
find the number of ways in which the game can be
won.
In an examination, the maximum marks for each
of three papers is n and that for the fourth paper is
2n. Prove that the number of ways in which a
candidate
can
get
3n
marks
is
1
(n  1)(5n 2  10n  6) .
6
7.
Two packs of 52 playing cards are shuffled together.
Find the number of ways in which a man can be
dealt 25 cards so that he does not get two cards of
the same suit and same denomination.
9.
There are m points on one straight line AB and n
points on another straight line AC, none of them
being A. How many triangles can be formed with
thest points as vertices ? How many can be fored if
point A is also included ?
10.
If n distinct things are arranged in a circle, show
that the number of ways of selecting three of these
things so that no two of them are next to each other
There are n straight lines in a plane, no two of which
are parallel and no three passes through the same
point. Their point of intersection are joined. Show
that the number of fresh lines thus introduced is
1
n (n  1)(n  2)(n  3)
8
6.
8.
Show that the number of ways of selecting n things
out of 3n things, of which n are of one kind and
alike, and n are of a second kind and alike and the
rest are unlike, is (n + 2)2n – 1
Einstein Classes,
is
1
n (n  4)(n  5) .
6
n
11.
Show that
(
n
C j )( j Ci )  ( n Ci )2 n 1 for i  n.
j i
12.
A box contains two white, three black and four red
balls. In how many ways can three balls be drawn
from the box if at least one black ball is to be
included in the draw ?
13.
Suppose a city has m parallel roads running
East-West and n parallel roads running
North-South. How many rectangles are formed with
their sides along these roads ? If the distance
between every consecutive pair of parallel roads is
the same, how many shortest possible routes are
there to go from one corner of the city to its diagonally opposite corner ?
14.
Find the number of all whole numbers formed on
the screen of a calculator which can be recognised
as numbers with (unique) correct digits when they
are read inverted. The greatest number formed on
its screen is 999999.
15.
In an examination, the maximum marks for each
of the three papers are 50 each. Maximum marks
for the fourth paper are 100. Find the number of
ways in which the candidates can score 60% marks
in the aggregate.
16.
n different things are arranged arround a circle. In
how many ways can 3 object be selected when no
two of the selected objects are consecutive.
17.
A dictionary is made of the words that can be made
by arranging the letters of the word PARKAR.
What is the position of the word ‘PARKAR’ in that
dictionary if words are printed in the same order
as that of an ordinary dictionary ?
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 12
18.
How many different rectangles are there on a
chessboard ? How many of these have the
area = 3 × area of small square ?
19.
From a panel of 5 lawyers, 5 chartered
accountants and 1 lawyer who is also a charactered
accountant, how many committees of four can be
made if each committee is to contain at least one
lawyer and one character accountant ?
20.
21.
Rekha married Shivram and had 4 sons. Varsha
married Ajoy and had 4 sons. Both the couples had
divorce and after that Shivram married Varsha
while Ajoy married Rekha. They too had 3 sons each
from their wedlocks. How many selection of
8 children can be made from the 14 children so that
each of them have equal number of sons in the
selection ?
There are 5 eligible Punjabi grooms of which 3 know
Bengali and 5 eligible Bengali grooms of which 2
know punjabi. There are 5 eligible Punjabi brides
and 5 eligible Bengali brides. If an eligible groom is
agreeable to marry a girl of his comminity or
knowing her language and brides have to choice,
in how many different ways 10 couples can be
formed ?
22.
A person has 32 teeth or less in the mouth. Prove
that the largest sample of people will contain 168
people in which no two persons have the same
setting of teeth.
23.
A batsman scores exactly a century by hitting fours
and sixes in twenty consecutive balls. In how many
different ways can he do it if some balls may not
yield runs and the order of boundaries and
overboundaries are taken into account ?
24.
Find the number of integral solutions of the
equation
x1 + x2 + x3 + .... + xk = 4k2 – 2k + 1
where x1  1, x2  3, x3  5,....,xk  2k – 1
25.
There are 4 pairs of hand gloves of 4 different
colours. In how many ways can they be paired off
so that a left-handed glove ahd a right-handed glove
are not of the same colour ?
26.
Find the number of selections of 10 balls from
unlimited number of red, black, white and green
balls. Also find how many of the selections contain
balls of the four colours.
ANSWERS (SINGLE CORRECT CHOICE TYPE)
Einstein Classes,
1.
c
14.
c
2.
a
15.
c
3.
a
16.
a
4.
c
17.
b
5.
b
18.
c
6.
c
19.
d
7.
d
20.
a
8.
a
21.
a
9.
c
22.
c
10.
a
23.
c
11.
b
24.
a
12.
c
25.
c
13.
a
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
MPC – 13
A N S W E R S (Subjective)
(Initial Step Exercise)
1.
70
2.
4373
3.
1001
9.
m  n!
mn
10.
m  n – 2!
m  1! n  1!
11.
n
(n  4)(n  5).
6
12.
(i)
(ii)
(n  3)(n 2  4)
6
13.
190
14.
26
15.
336
16.
25920
17.
98280
18.
2m
n –2
7.
4410
Cm – 1
A N S W E R S (Subjective)
(Final Step Exercise)
1.
4.
(p – 2)
C3
n
C2 × 3n – 2
2.
(4n2 – 6n + 4) (2n – 2) !
5.
96
8.
52
12.
14.
15.
16.
19.
64
13.
100843
110556
n/6 (n – 4) (n – 5)
320
20.
23.
1
1
1 
 1
20! 




 16!3! 14!4!2! 12!7! 10!10! 
26.
286, 84
Einstein Classes,
C26 . 226
9.
mn (m  n )
triangles
2
m+n–2
Cm – 1 ways
17.
485
99th
21.
24.
18.
1296, 96
144000
k(3k – 1)
Ck – 1
25.
9
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111