[8] COUNTING PRINCIPLES
1) State the Basic Multiplication Principle of Mathematics
2) A person wants to purchase a cellular phone and a calling plan. Suppose that
there are only two choices of cellular phones (the Motorola and the Nokia)
and three possible calling plans (one for $29.99 which allows 300 minutes of
airtime per month, a second for $39.99 which allows 600 minutes of airtime
per month, and a third for $49.99 which allows 1000 minutes of airtime per
month). In how many different ways can this person purchase a cellular
phone and a calling plan?
SOLUTION:
Number of different ways:
View the different possibilities by constructing a tree diagram or a matrix.
TREE DIAGRAM
300
M
600
1000
root
300
S
600
1000
MATRIX
Cell
M
S
POSSIBILITIES
(M,300)
(M,600)
(M,1000)
(S,300)
(S,600)
(S,1000)
300
600
1000
3) A K.G. activity consists of counting the number of squares in the rectangle
below. Use the multiplication principle to count the number of squares.
SOLUTION:
4) A game consists of tossing a coin once and then rolling a die once. How
many different outcomes are possible?
SOLUTION: Coin: {H, T}; n1 = 2; Die: {1, 2, 3, 4, 5, 6}; n2 = 6
Total = (2)(6) = 12 different possible outcomes.
We may view the 12 outcomes by constructing a matrix or a tree diagram.
MATRIX
DIE OUTCOMES
COIN
1
2
3
4
5
6
Heads (H)
(H,1)
(H,2) (H,3) (H,4) (H,5)
(H,6)
Tails (T)
(T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6)
TREE DIAGRAM
H
POSSIBLE OUTCOMES
1
(H, 1)
2
(H, 2)
3
(H, 3)
4
(H, 4)
5
(H, 5)
6
(H, 6)
root
T
1
2
3
4
5
6
(T, 1)
(T, 2)
(T, 3)
(T, 4)
(T, 5)
(T, 6)
Either way, we can view the 12 different possible outcomes.
5) A particular kind of code consists of two digits. Each digit comes from the set
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. How many different two-digit codes can be formed?
6) A candidate for governor needs a campaign director and a debate advisor. If
8 people apply for campaign director and 10 apply for debate advisor, in how
many different ways can the candidate select the persons that s/he needs?
7) State the General Multiplication Principle
8) At a particular restaurant, meals consist of one choice of {steak, chicken,
fish}, one choice of {baked potato, mashed potatoes}, and one choice of
{water, soda, juice}. In how many different ways can a customer order a
meal? Construct a tree diagram to illustrate all possible meals.
SOLUTION: (3)(2)(3) = 18 different meals
TREE DIAGRAM
POSSIBLE MEALS
baked
water
soda
juice
(steak, baked potato, water)
(steak, baked potato, soda)
(steak, baked potato, juice)
mashed
water
soda
juice
(steak, mashed potato, water)
(steak, mashed potato, soda)
(steak, mashed potato, juice)
baked
water
soda
juice
(chicken, baked potato, water)
(chicken, baked potato, soda)
(chicken, baked potato, juice)
mashed
water
soda
juice
(chicken, mashed potato, water)
(chicken, mashed potato, soda)
(chicken, mashed potato, juice)
baked
water
soda
juice
(fish, baked potato, water)
(fish, baked potato, soda)
(fish, baked potato, juice)
mashed
water
soda
juice
(fish, mashed potato, water)
(fish, mashed potato, soda)
(fish, mashed potato, juice)
steak
root chicken
fish
9) A restaurant offers five choices of appetizers, six choices of main courses,
seven choices of beverages, four choices of desserts. If exactly one item is
selected from each group, in how many ways can a person order a meal?
10) A certain model of vehicle is available in twelve different colors, four
different styles {hatchback, sedan, SUV, or station wagon}, {manual or
automatic transmission}, and {two-door or four-door}. How many different
possible choices of this vehicle are there?
11) A pizza can be ordered with four choices of size {small, medium, large,
super-large}, four choices of crust {thin, thick, crispy, regular}, and ten
choices of toppings {extra cheese, beef, chicken, ham, bacon, sausage,
pepperoni, mushrooms, onions, green peppers}. How many different onetopping pizzas can be ordered?
12) A true-false test consists of twelve questions. In how many different ways
can an unprepared student guess the twelve answers?
13) A social security number consists of nine digits. How many social security
numbers can be formed?
14) Five candidates are selected to fill in the positions of president, vicepresident, treasurer, secretary, and receptionist of a 200-member club. If
each candidate must be assigned to exactly one position and each position
must be assigned exactly one candidate, in how many different ways can the
positions be filled?
15) Six horses compete in a horse race. If there are no ties and all six horses
finish the race, in how many different ways can the race end?
16) Suppose that four textbooks must be placed in a bookshelf with four slots.
If you place each book in exactly one slot and each slot must contain exactly
one book, in how many different ways can you arrange the four textbooks?
17) FACTORIAL NUMBERS: In general, if n is a finite whole number, then
n! = n(n – 1)(n – 2) … (2)(1).
By definition: 0! = 1 and 1! = 1
Some factorial number computations are shown below.
0! = 1
1! = 1
2! = (2)(1) = 2
3! = (3)(2)(1) = 6
4! = (4)(3)(2)(1) = 24
5! = (5)(4)(3)(2)(1) = 120
6! = (6)(5)(4)(3)(2)(1) = 720
7! = (7)(6)(5)(4)(3)(2)(1) = 5040
8! = (8)(7)(6)(5)(4)(3)(2)(1) = 40,320
9! = (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880
10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3,628,800
11! = (11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 3,991,680
12! = (12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1) = 47,900,160
18) Interpretation of n!
19) PERMUTATIONS
20) Compute the number of permutations of all the elements of the set {A, B, C}.
List the permutations.
21) For the case of three candidates {Amal, Marie, Sy} running for the positions
of President, Vice-president, and Secretary of a particular club with exactly
one candidate for each position and exactly one position for each candidate,
how many permutations are possible? List all possible permutations.
President
Vice-President
Secretary
22) A swimming race consists of eight contestants. Assuming no ties, in how
many different ways can the race end?
23) One at a time, in how many different ways can 5 people lineup to get on a bus?
24) In how many different ways can seven people A, B, C, D, E, F and G sit in
a row at the opera?
25) MORE PERMUTATIONS
The number of permutations of the n elements of a set taken k at a time (k ≤ n)
is represented by the symbols
formula is
or P(n, k) or
and the computational
= n(n – 1)(n – 2) … (n – k + 1)
26) Suppose that ten candidates run for three offices: President, Vice-president,
and Secretary with exactly one person in each office and exactly one office
for each person. In how many different ways can the three offices be filled?
27) A particular competition awards $200,000 to the first place, $100,000 to the
second place, and $50,000 to the third place. If there are twelve competitors,
in how many different ways can the three prizes be awarded?
28) Evaluate: a)
29) IN GENERAL:
b)
c)
= n! and
d)
e)
f)
g)
= 1, for all finite integers n ≥ 0.
30) In how many ways can the eight members of a committee select a
chairman, a vice-chairman, and a minute-taker?
31) In how many can 18 Navy Officers select an Admiral, a Vice-Admiral, a
Rear Admiral Upper Half, a Rear Admiral Lower Half, and a Captain?
32) If a president, a vicepresident, a provost, a viceprovost, an academic dean,
and one assistant to the academic dean must be selected from a group of 16
university officials, in how many ways can this be done?
33) COMBINATIONS
34) COMPUTATIONAL FORMULAS
a) number of combinations of all the n elements of a finite set is 1.
b) number of combinations of the n elements of a finite set taken k at a time
(of course, k ≤ n) is denoted by
given by
or C(n, k) or
or
and
=
35) How many different subsets of two elements can be formed from the set
{A, B, C, D}? List the two-element subsets.
36) How many different subsets of three elements can be formed from the set
{A, B, C, D, E}? List the three-element subsets.
37) DUALITY :
38) Verify: a)
b)
c)
39) How many three-member committees can be formed from a group of ten
individuals?
40) A basketball team consists of 24 players who can play well in any position.
In how many ways can the coach select five players for a game?
41) If there are five nursing positions to be filled at a local hospital and 28
registered nurses apply for the jobs, in how many ways can the five nurses
be selected from the 28 applicants?
42) In the Florida Lotto game players select 6 numbers from {1, 2, 3, 4, …, 53}.
Any player who matches the six winning numbers in any order wins the
jackpot. If more than one player match the six winning numbers, then the
winners must divide the jackpot equally among themselves. In how many
ways can a player select six numbers to play the Florida Lotto?
43) In the Florida Lotto game, in how many ways can a player match three of
the six winning numbers?
44) A shipment of 40 plasma TV sets contains six damaged TVs. A sample of
10 TVs is selected for testing. a) How many samples of 10 are possible?
b) How many samples of 10 contain exactly three of the damaged TVs?
45) How many committees consisting of three women and three men can be
formed from a group of 23 individuals of which eight are men and the rest
are women?
46) Consider a group of 8 Democrats, 6 Republicans, and 5 Independents.
a) How many three-member committees can be formed if each party must
be represented? b) How many six-member committees can be formed if
two members of each party must be in the committee? c) How many
three-member committees consisting of one chairman, one vice-chairman,
and one minute-recorder, can be formed?
47) How many twelve-member committees consisting of five lawyers and
seven tax preparers can be formed from a set of people consisting of ten
lawyers and twelve tax preparers?
48) Consider the set of candidates {E, I, J, L, N, O, Q, R, T, U, V, Z}
a) How many subsets does this set have? b) In how many ways can a
voter place the candidates in a preference list ballot?
c) How many coalitions of seven candidates each can be formed?
49) A group of seven girls and four boys must sit in a straight row.
a) In how many ways can the eleven boys and girls sit in a straight row?
b) If the seven girls must sit together, in how many ways can the eleven
boys and girls sit in a straight row? c) If the seven girls must sit together
and the four boys must sit together too, in how many ways can the
eleven boys and girls sit in a straight row?
Applications of Counting to Probability
1) A group of professors consists of 3 from engineering, 4 from humanities, 5
from business, 4 from English, and 3 from the mathematics department.
a) Find the probability that a five-professor committee selected from this
group consists of exactly one professor from each mentioned department.
SOLUTION: P =
(3)(4)(5)(4)(3)
720
20
=
=
= 0.0619
11,628 323
19 C 5
b) Find the probability that a ten-professor committee selected from this
group consists of exactly two professors from each mentioned department.
SOLUTION:
=
= 0.0351
2) A basketball team consists of a total of 24 players, each of which plays very
well every position.
a) In how many ways can the coach select five players for a game?
SOLUTION:
= 42,504 ways
b) If Bill Gil is a team player, find the probability that Bill Gil will be among
the five selected. SOLUTION:
= 0.20833
3) A shipment of 40 plasma TV sets contain 4 defectives. A customer purchases
6 sets from this shipment. Find the probability that s/he gets:
a) No defective sets.
SOLUTION:
= 0.50745
b) Exactly one defective set.
SOLUTION:
c) At most one defective set.
SOLUTION:
= 0.39287
= 0.90032
d) At least one defective set.
SOLUTION:
= 0.49255
Also: P = 1 – P(0) = 1 – 0.50745 = 0.49255
e) Two defective sets.
SOLUTION:
= 0.09208
f) All the defective sets.
SOLUTION:
= 0.0001641
4) Find the probability that a 7-card poker hand contains 2 aces and 3 face cards.
SOLUTION:
5) A box contains twelve books, among which there is one history book, one
math book, one literature book, and one biology book. If four books are
taken from this box, at random, one by one, without replacement, find the
probability that the four mentioned books are selected in the particular order
listed above.
SOLUTION:
6) In the Florida Lotto game players select 6 numbers from {1, 2, 3, 4, …, 53}.
A player wins the jackpot if s/he matches the six winning numbers in any
order. If k players match the 6 winning numbers, they must divide equally
the jackpot among themselves.
a) Find the probability of winning the jackpot.
b) Find the probability of matching 3 winning numbers.
EXERCISES:
1] A secret code consists of one digit followed by one letter from the Alphabet.
How many different codes of this kind can be formed?
2] A woman has 12 different skirts and 16 different blouses. How many
different outfits can she wear?
3] A restaurant offers nine appetizers and twelve main courses. How many
different meals consisting of one appetizer and one main course can a
customer order?
4] A man has 5 shirts, 4 ties, 6 pairs of pants, 8 pairs of socks, and 3 pairs of
shoes. Find the number of different outfits that he can form if an outfit
includes one item from each group of items.
5] A woman has 5 blouses, 8 skirts, 20 pairs of shoes, and 10 purses. If an outfit
includes exactly one item from each group of items, how many different
outfits can she wear?
6] How many 4-digit numbers can be formed from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
if the first digit cannot be zero and digits cannot be repeated.
7] A bank security system requires customers to choose a 5-digit personal
identification number (PIN). If all digits must be different (no repetitions
allowed) and nonzero, how many different five-digit PINs can be formed?
8] How many (three-digit) area codes exist if the first digit cannot be zero or
one, the second digit must be zero or one, (no restriction on the third digit)?
9] A true-false exam consists of six questions. If an unprepared student takes
this exam, in how many ways can the student answer the six questions?
10] A multiple-choice exam consists of six questions, each question having
options a, b, c, d, with only one option being correct. If an unprepared student
takes this exam, in how many ways can s/he answer the six questions?
11] Six performers are to present their comedy acts on a weekend evening at a
comedy club. If the acts are presented one by one, in how many different
ways can the performers schedule their appearances?
12] In how many different ways can nine books be arranged in a bookshelf?
13] Ten singers must perform on a weekend evening at a night club. How many
different ways are there to schedule their appearances?
14] In how many different ways can a police department arrange eight suspects
in a police lineup if each lineup contains all eight people?
15] 4 candidates must be assigned to the positions of President, Vice-president,
Treasurer, and Secretary. Each office must be assumed by exactly one
person and each person must be assigned exactly one office. In how many
different ways can this be done?
16] There are three routes from College Park to Baltimore and five routes from
Baltimore to New York. How many routes are there from College Park to
New York?
17] How many different outfits consisting of a coat and a hat can be chosen
from six coats and four hats?
18] How many different license plates consisting of three letters followed by
three digits are possible?
19] The Spoiled & Rotten Restaurant offers 8 choices of entrees, 5 choices of
salads, 10 choices of beverages, and 4 choices of desserts. How many
different meals can a customer order?
20] a) In how many ways can six people be arranged in a line to take a picture?
b) In how many ways can they be arranged if three of the six insist on
appearing one next to one another?
c) In how many ways can they be arranged if two of the six refuse to appear
one next to the other?
21] In how many ways can five books be arranged on a bookshelf?
22] How many rearrangements of the letters L, S, A, B are possible? List all
possible rearrangements.
23] Twelve participants enter an Olympic event. In how many ways can they be
awarded the Gold Medal to the first place, the Silver Medal to the second
place, and the Bronze Medal to the third place?
24] In how many ways can a 24-member team select a captain and an assistant?
25] 12 investors want to elect a president, a vice-president, a secretary, and a
treasurer from the twelve members. In how many ways can this be done?
26] In a particular state, one type of license plate consists of three uppercase
letters followed by three digits. (a) If repetitions are allowed, how many
license plates can be formed? (b) If repetitions are not allowed, how many
license plates can be formed?
27] A restaurant offers four soups, twelve entrees, nine beverages, and nine
desserts. In how many ways can a customer order a meal if three of the
desserts are pies and customers never order pies?
28] A group of 12 investors wants to form a needs a four-person committee.
How many four-person committees can be formed from the 12 members?
29] A group of 12 people consists of 5 Republicans and the rest are Democrats.
A five-person committee must be formed from these 12 individuals.
a) How many five-person committees can be formed?
b) How many of these committees consist of two Republicans?
c) How many of these committees consist of at least two Republicans?
d) How many of these committees consist of at most two Republicans?
30] In the Florida Lotto game players choose 6 numbers in any order from the set
{1, 2, 3, …, 53}. Any player who matches the six winning numbers in any
order wins the jackpot. a) How many different choices of six numbers exist?
b) In how many ways can a player match three of the six winning numbers?
31] A group of five girls and three boys must sit on a straight row. In how many
ways can they sit if: a) no restrictions are imposed? b) the three boys must
sit together one next to the other? c) the girls must sit together (one next to
the other) and the boys too.
32] A group of nine faculty members consists of 3 members from the English
department, 4 from Business, and the rest from Mathematics. A threemember committee must be formed from the group of 9. Find the number
of three-member committees if: a) no restrictions are imposed; b) each
department must have a representative; c) no restrictions are imposed
except that one member must be the chairperson, another member must be
the vice-chair, and the third must be the note-taker.
33] 6 performers will present their acts on a weekend evening at a comedy club.
How many different ways are there to schedule their appearances?
34] A pizza can be ordered with three choices of size (small, medium, or
large), four choices of crust (thin, thick, crispy, or regular), and six choices
of toppings (ground beef, sausage, pepperoni, bacon, mushrooms, or
onions). How many one-topping pizzas can be ordered?
35] A medical researcher needs 6 people to test a new drug. If 16 people have
volunteered for the test, in how many ways can 6 people be selected?
36] A multiple-choice test contains six questions, each having 4 options, of which
exactly one correct. In how many ways can a student answer the questions?
37] If three-digit codes are formed from the set {0, 3, 4, 5, 6, 7},
a) how many codes can be formed if repetitions are allowed?
b) how many codes can be formed if no repetitions are allowed?
c) how codes are three-digit odd numbers?
d) how many codes are three-digit numbers greater than 414?
38] In how many different ways can a police department arrange eight suspects
in a police lineup if each lineup contains all eight people?
39] A club with 10 members must choose 4 officers – president, vice-president,
secretary, and treasurer. If each office is to be held by exactly one person
and no person can hold more than one office, in how many ways can those
offices be filled?
40] In how many ways can a committee of five women and four men be formed
from a group of 18 people in which ten are women and the rest are men?
41] In a medical study, patients are classified by blood type and blood pressure.
If the blood types include {AB+, AB−, A+, A−, B+, B−, O+, O−} and the blood
pressures could be {normal, low, high}, find the number of ways in which a
patient can be classified.
42] a) How many permutations can be made from the letters of the word exam?
b) How many of these permutations end with the letter x?
43] a) A voter has to rank 4 candidates. How many different preference list
ballots can s/he make? b) If there are seven candidates in an election, how
many different preference list ballots can each voter make?
44] Simplify each of the following:
a) 8!
b)
16 !
12 !
c)
10 !
4!6!
d)
14 !
(14 − 4) !
e)
45] How many permutations are there of the letters ABCDE?
12 !
(12 − 4) ! 4 !
46] Evaluate: a)
b)
c)
d)
47] Prove that
48] How many subsets of size 4 can be formed from a set with 10 elements?
49] a) From a committee of 8 members, a 3-person subcommittee must be
formed. In how many different ways can this be done?
b) From a committee of 8 members, 3 must be chosen for the positions of
chair, vice-chair, and recording secretary. In how many different ways
can this be done?
COUNTING AND PROBABILITY
1) A group of individuals includes eight Democrats, six Republicans, and three
Independents. a) Find the probability that a six-person committee selected
from this group consists of exactly two of each political party. b) Find the
probability that a three-professor committee selected from this group consists
of exactly one of each political party.
2) An accounting firm consists of twenty employees and one manager (the
owner). The twenty employees want to form a four-person committee to issue
a complaint to the manager. a) How many four-person committees can be
formed from the 20 employees? b) If Sally Gil is one the employees of this
firm, find the probability that Sally Gill will be among the four selected for
the committee.
3) A box containing 16 cellular phones includes 3 defectives. A sample of 5
phones is selected are random from this box. Find the probability that the
sample includes: a) No defective phones; b) Exactly one defective phone;
c) At most one defective phone;
d) At least one defective phone;
e) Two defective phones;
f) All the defective phones.
4) a) Find the probability that a 5-card poker-hand contains 2 aces and 3 face cards
b) Find the probability that a 5-card poker hand contains exactly 2 aces.
c) Find the probability that a 5-card poker hand contains exactly 3 face cards.
5) A 3-digit code is selected from the set of digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. If
no repetitions are allowed, find the probability that the code 0-5-9 is selected.
6) In the Fantasy-5 game, players choose 5 numbers from the set {1, 2, 3, 4, …, 36}.
A player wins the jackpot if s/he matches the five winning numbers in any
order. If k players match the five winning numbers in any order, they must
divide equally the jackpot among themselves. a) Find the probability of
winning the jackpot. b) Find the probability of matching three of the five
winning numbers.
ANSWERS
[1] (10)(26) = 260 codes; [2] (12)(16) = 192 outfits; [3] (9)(12) = 108 meals
[4] (5)(4)(6)(8)(3) = 2880 outfits; [5] (5)(8)(20)(10) = 8000 outfits;
[6] (9)(9)(8)(7) = 4536 four-digit numbers; [7] (9)(8)(7) = 504 PINs;
6
6
[8] (8)(2)(10) = 160 area codes; [9] 2 = 64 ways; [10] 4 = 4096 ways
[11] 6! = 720 ways; [12] 9! = 362,880 ways; [13] 10! = 3,628,800 ways
[14] 8! = 40,320 ways; [15] 4! = 24 ways; [16] (3)(5) = 15 routes;
[17] (6)(4) = 24 outfits; [18] (26)(26)(26)(10)(10)(10) = 17,576,000 plates
[19] (8)(5)(10)(4) = 1600 meals; [20] a) 6! = 720 ways; b) (4!)(3!) = 144
ways; c) 6! – (5!)(2!) = 480 ways; [21] 5! = 120 ways;
[22] 4! = 24 ways; LSAB, LSBA, LASB, LABS, LBSA, LBAS, SLAB, SLBA, SALB,
SABL, SBLA, SBAL, ALSB, ALBS, ASLB, ASBL, ABLS, ABSL, BLSA, BLAS,
BSLA, BSAL, BALS, BASL;
[23] 12 P3 = 1320 ways; [24] 24 P2 = 552 ways; [25] 12 P4 = 11,880 ways;
[26] a) (26)(26)(26)(10)(10)(10) = 17,576,000 plates;
b) (26)(25)(24)(10)(9)(8) = 11,232,000 plates;
[27] (4)(12)(9)(6) = 2592 meals; [28] 12 C 4 = 495 committees;
[29] a) 12 C 5 = 792 five-person committees; b) ( 5 C 2 )( 7 C 3 ) = 350 committees;
c) ( 5 C 2 )( 7 C 3 ) + ( 5 C 3 )( 7 C 2 ) + ( 5 C 4 )( 7 C1 ) + ( 5 C 5 )( 7 C 0 ) = 596 committees;
( 5 C 0 )( 7 C 5 ) + ( 5 C1 )( 7 C 4 ) + ( 5 C 2 )( 7 C 3 ) = 546 committees;
a) 53 C 6 = 22,957,480 possibilities; b) ( 6 C 3 )( 47 C 3 ) = 324,300 ways
d)
[30]
[31] a) 8! = 40,320 ways; b) (6!)(3!) = 4,320 ways; c) (2)(5!)(3!) = 1440 ways
[32] a) 9 C 3 = 84 ways; b) (3)(4)(2) = 24 ways; c) 9 P3 = 504 ways
[33] 6! = 720 ways; [34] (3)(4)(6) = 72 one-topping pizzas; [35] 16 C 6 = 8008
6
[36] 4 = 4096 ways; [37] a) (6)(6)(6) = 216 codes; b) (6)(5)(4) = 120 codes;
c) (6)(6)(3) = 108; d) (1)(5)(6) + (3)(6)(6) = 138; [38] 8! = 40,320 ways;
[39] 10 P4 = 5040 ways; [40] ( 10 C 5 )( 8 C 4 ) = 17,640 ways; [41] (8)(3) = 24 ways;
[42] a) 4! = 24; b) (3)(2)(1)(1) = 6; [43] a) 4! = 24 ballots; b) 7! = 5040 ballots
[44] a) 40,320; b) 43,680; c) 210; d) 24,024; e) 495; [45] 5! = 120 permutations
[46] a) 5040; b) 210; c) 30; d) 15; [48] 10 C 4 = 210; [49] a) 8 C 3 = 56; b) 8 P3 = 336 ways.
Answers to Counting and Probability Exercises
1) a)
2) a)
3) a)
; b)
; b)
; b)
d)
; c)
= 1 – P(0); e)
f)
4) a)
; b)
5)
6) a) 0.000002652; b) 0.01233
; c)
;
;