Permutations
and
Combinations
6.7
Bell Work
• Without looking at your notes, find the 5th
term
(2 x  3 y)
6
2
4860x y
4
Counting Principle
• In how many ways can 5 kids line up for
play time?
5  4  3  2 1  120 ways
Factorial
5! = 5  4  3  2  1
20! = 20  19  18  17  16  ...  1
n! = n(n  1)  ...3  2 1
Sample Problems
1. How many 4 digit PIN codes are possible using only
numbers?
10 10 10 10  10,000
2. How many 4 digit PIN codes are possible using
numbers and letters?
36  36  36  36  1,679,616
3. How many 6 digit license plates are possible if the first 3
digits are numbers and the last 3 digits are letters, but
none of the numbers can repeat?
(10  9  8)  (26  26  26)  12,654,720
“Picky” Permutations
A permutation is an arrangement of
items in a particular order.
Sample Permutations
1. In how many ways can you arrange
six trophies on a shelf?
6!  720
2. Seven boats enter a race. How
many arrangements of first, second,
and third are possible?
7  6  5  210
3. There are ten players on a basketball team.
In how many ways can a starting lineup of five
players be chosen?
P  10  9  8  7  6  30240
10 5
4. Three positions (President, Vice-president,
and Secretary) must be filled out of 15
applicants. How many ways are possible?
P  15 14 13  2730
15 3
5. How many 4 digit PIN codes are possible
using only numbers, if no number can be used
more than once?
P  10  9  8  7  5040
10 4
6. How many ways can the letters of the word
DERF be arranged?
P  4  3  2 1  24
4 4
Combinations (Clusters)
A combination is a selection in
which order does not matter.
Sample Combinations
1. How many three person committees
can be formed out of five people?
5
C3  10
2. Seven boats enter a race. The first
three boats will all win equal prizes.
How many different combinations of
winners are possible?
7
C3  35
3. Fifty boys are trying out for soccer. Only
twenty-five of them will make the team. How
many teams are possible? C  1.26411014
50
25
4. Ten candidates are running for three seats in
the student government. How many groups of
three are possible?
10
C3  120
5. A pizzeria offers 10 different toppings. In
how many ways can you choose four toppings?
10
C4  210
Homework
Worksheet
Pg. 342 #11-31 odd, 46-49 all