Introductory Statistics

Lesson 3.4 A

Objective:

SSBAT determine the number of permutations.

Standards: M11.E.3.2.1

Permutation

An arrangement of data where order is IMPORTANT

ABC is not the same as CBA

Can use the Fundamental Counting Principle to find

Can also find the number of different permutations of n distinct objects using n!

n!

Read as n factorial

Defined as: n! = n(n – 1)(n – 2) (n – 3) ···3·2·1

Examples:

5!

9!

= 5 · 4 · 3 · 2 · 1 = 120

= 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 362,880

There is a factorial key on the calculator

Go to MATH Scroll over to PRB Then choose #4 which is !

Practice: 8!

12!

10! – 4!

= 40,320

= 479,001,600

= 3,628,776

Example: Find the number of ways 6 people can finish a race.

Method 1: Fundamental Counting Principle

6 · 5 · 4 · 3 · 2 · 1 = 720

Method 2: Permutation

6! = 720

Permutations of n Objects taken r at a time

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

nPr =

n is the total objects you have

r is how many you are picking

Find each of the following. nPr =

7P4

24P11

Answer: 840

Answer: 99,638,080,820,000

Examples:

1. Forty-three race cars started the Daytona 500 in 2007. How many ways can the cars finish first, second and third?

n = 43 and r = 3

43P3 =

43P3 =

43P3 = 74,046

2. The board of directors for a company has 12 members. One member is the president, another is the vice-president, another is the secretary, and another is the treasurer. How many ways can these positions be assigned?

n = 12 and r = 4

12P4 =

12P4 =

12P4 = 11,880

3. A museum is going to hang 9 paintings on a wall from left to right. How many different ways can the museum hang these 9 paintings on the wall?

n = 9 and r = 9

9P9 =

9P9 = 362,880

Distinguishable Permutations

The number of distinguishable permutations of n objects, where n1 are of one type, n2 are of another type, and so on is

(where n1+ n2 + n3 + …nk = n)

Example: A building contractor is planning to develop a subdivision with a total of 12 houses. The subdivision is to consist of 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged?

There are 12 houses in the subdivision, 6 are one type, 4 are another type, and 2 are a third type.

13,860 distinguishable ways

Complete Worksheet 3.4 A