Warm Up 1/31/11

1. If you were to throw a dart at the purple area, what would be the probability of hitting it?

13

20

I-------8-------I

5

13

20

I-------8-------I

5

Area of Blue = 20·13

A = 260

Area of purple = ½ (8)(3)

A = 123

Probability of hitting purple = 12/260

P = 3/65

15.2

Objective: To use the FUNDAMENTAL COUNTING PRINCIPLE and PERMUTATIONS to find the

possible number of arrangements

Remember…6! = 6·5·4·3·2·1

= 720

This is called a FACTORIAL

!3

!7Examples:

1) 5! = 5·4·3·2·1

= 120

123

1234567

2)

= 840

A. Arrangements

Example 1: How many possible 3-letter arrangements can be made using the 26 letters of the

alphabet? (repetition is allowed)

___ ___ ___

We can find the total number by multiplying all 3 together…

26·26·26 = 17, 576

This is called the FUNDAMENTAL COUNTING PRINCIPLE, which allows us to multiply together the possible outcomes for a series of events.

Example 2: How many 7-digit phone numbers can be created using 0-9?

(Restriction: the first 2 #’s can NOT be 0 or 1)

___ ___ ___ ___ ___ ___ ___

Total possibilities = 6,400,000

B. Permutations

There is a special type of arrangement called a PERMUTATION:

*repetition IS NOT allowed

*the order is important

Example 1: How many 4-letter permutations can be made using the letters A, B, C and D?

___ ___ ___ ___

Example 2: Brad is creating a 7 character screen name. The first 3 characters must be a letter from his name, and the last 4 characters must be a digit from the year 1987. How many different permutations are there?

___ ___ ___ ___ ___ ___ ___

Example 3: How many 5-letter permutations can be made using the letters in the word “FISHER”?

___ ___ ___ ___ ___

Another way this can be written is: 56P

Total # of itemsThe # we want

In General:

)!(

!

rn

nPrn

Calculate the following:

1) 2) 3) 25P 36P

)!25(

!5

123

12345

20

)!36(

!6

123

123456

120

27P

C. Permutations with repeating letters:

If there are repeating letters in a word with n total letters, to find the number of permutations we use:

!...!!

!

321 rrr

n

Where represent the number of times that a letter repeats itself.

...,, 321 rrr

Example 1

How many 7-letter permutations can be made from the letters in the word “CLASSIC” ?

!2!2

!7

1212

1234567

1260

Example 2

How many 11-letter permutations can be made from the letters in MISSISSIPPI?