Fundamental Counting Principle, Permutations and Combinations

Fundamental Counting Principle

The number of ways in which a series of

successive things can occur is found by

multiplying the number of ways in which each

thing can occur.

Sally is packing for vacation. She wants to maximize her

wardrobe options. She plans on taking the following clothing

items. How many possible outfits will she have on her trip?

jeans 2 pairs, shorts – 3 pairs, skirts – 2, shirts – 9, shoes – 4 pairs

7 bottoms

Number of Outfits =

Example 1

252 7 • 9 • 4 =

Example 2:

You are taking a multiple choice test

that has five questions. Each of

the questions has five possible

answers with one correct answer

per question. If you select one

answer per question how many

ways can you answer the

questions?

Q1 Q2 Q3 Q4 Q5

5 • 5 • 5 • 5 • 5 = 55

Example 3

Telephone numbers in the U.S.

begin with a 3 digit area code

followed by a seven digit local

number. Area codes and local

numbers cannot begin with a 0 or

1. How many different telephone

numbers are possible?

Possible numbers per digit

8 • 10 • 10 • 8 • 10 • 10 • 10 • 10 • 10 • 10 = 6,400,000,000

area code local number

8 8 10 10 10 10 10 10 10 10

Permutations

An ordered arrangement that occurs when….

No item is used more than once.

The order of the arrangement makes a difference.

Example 4

You are the coach of a little league baseball team with 13 players.

You need to put together the batting order with nine players. How

many batting orders are possible?

Bat1 Bat 2 Bat 3 Bat 4 Bat 5 Bat 6 Bat 7 Bat 8 Bat 9

13

choices

11

choices

10

choices

9

choices

8

choices

7

choices

6

choices

5

choices

13 •12 • 11•10 • 9 • 8 • 7 • 6 • 5 = 259,459,200

12

choices

Permutations of n Things Taken r at a Time

The number of possible permutations if r items are taken

from n items is:

nPr = ______

n!

(n-r)!

Lets go back to our baseball batting order n = 13 r = 3

13P9 = _____ = __________________________ = 259, 459, 200

13! 13121110987654!

4! 4!

Example 5

A corporation has seven members on its board of

directors . In how many different ways can it select a

president, vice-president, secretary and treasurer?

840

Combinations

An arrangement that occurs when….

The items are selected from the same group.

No item is used more than once.

The order of the items makes no difference.

Distinguishing between Permutations & Combinations

Do you see the difference between a

permutation and a combination?

A permutation involves situations in which

order matters.

A combination involves situations in which

the order of the items makes no difference.

Example 6: Distinguishing between Permutations and Combinations

1. Six Students are running for student council president, vice-president

an treasurer. The student with the greatest number of votes becomes

the president, the student with the second greatest number of votes

becomes the vice-president and the student with the third greatest

number becomes the treasurer. How many different outcomes are

possible for these three positions?

Permutation or Combination

2. Six people are on the board of supervisors for the park district. A three

person committee is needed to study the possibility of expanding the

park. How many different committees could be formed from the 6

people?

Permutation or Combination

Example 6: Distinguishing between Permutations and Combinations

3. Baskin Robbins offers 31 different flavors of ice cream. One of their

items is a bowl consisting of three scoops of ice cream, each a

different flavor. How many such bowls are possible?

Permutation or Combination

4. In a race in which there are 50 runners and no ties, in how many

different ways can the first three finishers come in?

Permutation or Combination

Combinations of n Things Taken r at a Time

The number of possible combinations if r items are taken

from n items is:

ncr = ______

n!

(n-r)!r!

Example 7

From a group of 10 physicians , how many

ways can 4 doctors be selected to attend a

conference on accupuncture?

210

Example 8:

A person playing poker is dealt a 5 card hand from a

standard deck of 52 cards. The order in which the 5

cards are dealt does not matter. How many

different 5 card poker hands are possible?

= 2,598,960