Transcript Warm Up #4

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?
20
13
5
I-------8-------I
20
Area of Blue = 20·13
13
5
A = 260
Area of purple = ½ (8)(3)
3
A = 12
I-------8-------I
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
Examples:
1)
5! = 5·4·3·2·1
2)
7! 7  6  5  4  3  2  1

3  2 1
3!
= 120
= 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:
6
P5
Total # of items
The # we want
In General:
n!
n Pr 
(n  r )!
Calculate the following:
1)
5 P2
2)
P
6 3
5!
(5  2)!
6!
(6  3)!
5  4  3  2 1
3  2 1
6  5  4  3  2 1
3  2 1
20
120
3)
7
P2
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:
n!
r1! r2 ! r3 !...
Where r1, r2 , r3 ... represent the number
of times that a letter repeats itself.
Example 1
How many 7-letter permutations can be made from
the letters in the word “CLASSIC” ?
7! 7  6  5  4  3  2  1
 1260

2  1 2  1
2!2!
Example 2
How many 11-letter permutations can be
made from the letters in MISSISSIPPI?