7-4: Permutations and Combinations

Objectives:

1. Permutation 2. Combination 3. application

Defn: For a natural number n, we define the factorial, as follows:

12)2)(1(! ⋅⋅⋅⋅−−= nnnn

Note: )!1(! ,1!0 −== nnn

Ex:

Defn: A permutation of a set of distinct objects is an arrangement of the objects in a specific order without repetition.

Ex. how many ways can you arrange 5 books on a single shelf?

Theorem. The number of permutations of n distinct objects without repetition, denoted by nnP , , is

!12...)2)(1(, nnnnP nn =⋅⋅⋅−−=

A permutation of n objects taken r at a time without repetition is an arrangement of r of the n objects in a specific order.

Ex. how many ways can you arrange 3 out of the 5 books on a single shelf?

Theorem. The number of permutations of n objects taken r at a time is:

)1(...)2)(1(, +−⋅⋅−−= rnnnnP rn , or )!(

!, rn

nP rn −=

Ex: 2,5P

0,5P

5,5P

Ex. Given the set of objects {A, B, C, D}, how many permutations are possible for this set of 4 objects taken 2 at a time?

1) Solve using a tree diagram.

2) Solve using multiplication principle:

3) Solve using permutation formula rnP , :

Combinations

A combination of n objects taken r at a time without repetition is an r-element subset of the set of n objects. The arrangement of the elements in the subset does not matter.

Ex: The library wants to borrow any 2 of your 3 books. How many ways can you select these 2 books?

Theorem. The number of combinations of n distinct objects taken r at a time without repetition is given

by )!(!

!!,

, rnrn

rP

rn

C rnrn −

==

=

Ex:

7,12C

0,5C 5,5C

Ex. In how many ways can a four-member committee be formed from a group of 17 people?

Ex. How many 5-card hands have 3 hearts and 2 spades?

How do we know when to use permutation and when to use combination?

Permutation: order does matter;

Combination: order does NOT matter.

Ex. From a committee of 10 people.

A) In how many ways can we choose a chair person, a vice-chair person, and a secretary, assuming that one person cannot hold more than one position?

B) In how many ways can we select a subcommittee of 3 people?

Ex. A combination lock has 5 wheels, each labeled with the 10 digits from 0 to 9. How many 5-digit opening combinations are possible if no digit is repeated?

Ex: An office supply store receives a shipment of 24 high speed printers, including 5 that are defective. Three of these printers are selected for a store display.

a) How many selections can be made?

b) How many of these selections will contain no defective printers?

Ex. Suppose that 6 female and 5 male applicants have been successfully screened for 5 positions. In how many ways can the following compositions be selected?

A) 3 females and 2 males;

B) 4 females and 1 male;

C) 5 females;

D) 5 people regardless of sex;

E) At least 4 females.

Practice problems:

Find the number of permutations of 30 objects taken 4 at a time. Compute the answer using a calculator.

1. Find the number of combinations of 30 objects taken 4 at a time. Compute the answer using a calculator.

2. From a committee of 12 people.

a) In how many can we choose a chair person, a vice-chair person, a secretary and a treasurer, assuming that one person cannot hold more than one position?

b) In how many ways can we select a subcommittee of 4 people?

Answers to practice problems:

1. 657,720

2. a. 11,880 b. 495