MGF1106
12.8 Counting Principle and Permutations
COUNTING PRINCIPLE
If the 1st experiment can be performed M distinct ways & the 2nd experiment(independent) can be performed
in N distinct ways, then the 2 experiments in that order, can be performed M*N distinct ways(in that order).
****Must be used if specific placement or with replacement**
Ex1) License Plate = 3 letters and 3 digits in that order. How many ways can this be done if:
a) Repetition is allowed
b) No repetition, no 0 in the 1st digit
c) 1st letter has to be a vowel, 2nd letter is a consonant, no digits can be a 0, no repetition
Ex 2) you have 6 pictures of different people to hang on the wall in a row.
a) How many ways can you arrange this?
b) Suppose the 3rd and 4th position decided to be 2 specific people in any order.
How many ways can this be done?
Permutations

Any ORDERED arrangement of a given set of objects

Assume repetition is NOT allowed

Formula:
Number of Permutations of n distinct items (selecting all objects)
Ex 3) Only part of the items to be arranged
Vowels: {a, e, i, o, u}
How many ways can you arrange 3 vowels with no repetition?
NOTATION:
_______ = # of permutations of 5 items taken 3 at a time.
FORMULA *** # of Permutations when selecting PART of objects
when r objects are selected from n objects
n = Total # of objects
r= # objects selected
Ex4) There are 10 members on a basketball team. The coach is going to randomly select a captain,
1st alternate captain, 2nd alternate captain. How many choices does he have?
Permutations of DUPLICATE items
Ex 5) How many permutations of letters in the word DAD?
FORMULA *** Permutations of DUPLICATE items
n=# of objects
n1= objects the same
n2= objects the same
n3= objects the same
nr= objects the same
Ex 6) How many permutations of the letters in the words Mississippi?
Permutations:
1) n!
2) n Pr 
n!
(n  r )!
3) repetition:
n!
(n1 )!(n2 )!...(nr )!
Examples from book (pg 724):
22. There are three different routes that Cassidy can walk from home to the post office and two
different routes that she can walk from the post office to the bank. How many different
routes can Cassidy walk from home to the post office and then to the bank?
24. Assume that a password to log onto a computer account is to consist of any four digits or
letters (repetition is permitted). Determine the number of passwords possible if:
a) the letters are not case sensitive
b) the letters are case sensitive
28. The trifecta at most racetracks consists of selecting the first, second, and third place winners
in a particular race in their proper order. If there are seven entries in the trifecta race, how
many tickets must you purchase to guarantee a win?
36. A DJ has 9 songs to play. Five are slow songs, and 4 are fast songs. Each song is to be
played only once. In how many ways can the DJ play the 9 songs if
a) the songs can be played in any order?
b) the first song must be a slow song and the last song must be a slow song
c) the first two songs must be fast songs?
46. Mrs. Hernandez and her three children go shopping at a local grocery store. Each of the
children will be allowed to select one bottle of fruit juice. On the store’s shelf there are 12
bottles of fruit juice, and each bottle contains a different type of juice. In how many ways
can the selection be made?
54. Determine the number of permutations of the letter of the word “EFFECTIVE.”
12.9 COMBINATIONS
n
Cr 
n!
(n  r )!(r )!
.
Combination

Distinct set of objects WITHOUT regard to arrangement

repetition not allowed
Permutations: order matters!
Combinations: order doesn’t matter to outcome
Neither allow for repetition
EX1) Permutations or Combinations
5 starters on a basketball team
a) How many ways can you choose a captain and alternate captain?
b) How many ways can you choose 2 starters to discuss topics with the coach?
.
FORMULA *** # of COMBINATIONS of n items taken r at a time
NOTATION: _______ = # of combinations of n items taken r at a time
Ex 2) How many ways can you select 2 different vowels where order doesn’t matter?
Ex 3)Pres. Mojock invites 10 faculty members to lunch. There is room for 6 faculty members in his car.
In how many ways can 6 faculty members be chosen to ride with Pres. Mojock?
Ex 4) You have a test on which there are 6 questions. You are required to select any 4 questions which
are worth the same amount of points. How many different tests are available to you?
Ex 5) Priceless Pizza is offering a special on a 3 topping pizza. They have 8 different toppings
available. How many different ways can you order a pizza with different toppings?
Ex 6) A caterer offers a choice of 6 meats, 8 vegetables, and 5 desserts. For your party, you wish to
have 2 meats, 3 vegetables, and 2 desserts. How many different selections are available to you?
Pg 749
22. An ice-cream parlor has 20 different flavors. Cynthia orders a banana split and has to select 3
different flavors. How many different selections are possible?
24. During a special promotion at CompUSA, a customer purchasing a computer and a printer is given
the choice of 2 free software packages. If there are 9 different software packages from which to select,
how many different ways can 2 packages be selected?
26. While visiting NYC, the Nygens want to attend 3 plays out of 10 plays they would like to see. In
how many ways can they do so?
28.Mary Robinson purchased a package of 24 different plants, but she only needed 20 plants for
planting. In how many ways can she select the 20 plants from the package to be planted?
30. James has 7 CDs and wants to select 4 to play in his car CD player. In how many ways can he
select 4 of 7 to play?
34. On an English test, Tito must write an essay for three of the five questions in Part 1 and four of the
six questions in Part 2. How many different combinations of questions can he answer?
36. An animal shelter has 15 cats and 12 dogs available for adoption. The shelter wants to select 6 cats
and 4 dogs to feature in the adoption newsletter. In how many ways can this be done?
42. As part of a door prize, Mary won three tickets to a baseball game and three tickets to a theater
performance. She decided to give all the tickets to friends. For the baseball game she is considering six
different friends, and for the theater she is considering eight different friends. In how many ways can
she distribute the tickets?