Combinations and Permutations
October 13, 2009
Objectives
 Content
 Students will learn about permutations and combinations.
 Language
 Students will participate in a orderly manner.
Combinations vs. Permutations
 If the order doesn't matter, it is a Combination.
 Combinations of fruits in a salad
 Combinations of people on a team
 If the order does matter it is a Permutation.
 Ways to line up a group of people
Permutations
 A permutation is an ordering or arrangement.
 If there can be repetition, this is exactly what we have been
doing already.
 If you have n things to choose from, and you choose r of
them, then the permutations are:
n × n × ... (r times) = nr
(Because there are n possibilities for the first choice, THEN
there are n possibilites for the second choice, and so on.)
 For example in the lock above, there are 10 numbers to
choose from (0,1,..9) and you choose 3 of them:
 10 × 10 × ... (3 times) = 103 = 1000 permutations
An example
 Suppose that there are five students who need to ask for help
on their Algebra homework. How many different orders
could there be?
Another example
 There are 15 students in speech. How many arrangements
are there for 3 students to give a speech?
Permutations
 P(n, r) represents the number of permutations for n
elements, taken r at a time.
 ! means factorial.
 Examples:
3!=3*2*1=6
0!=1
1!=1
P(n, r)
 P (n, r) =
n!
(n  r )!
 n=number of elements
 r=the number taken at a time
Example
 How many ways two letter strings can be formed from A, B,
C, and D?
Combinations
 Order DOES NOT matter.
 How many teams of two can be created from a selection of
five players?
C(n,r)
 C(n,r) =
n!
r!(n  r )!
 n=number of elements
 r=the number taken at a time

An example
 You have five courses left in high school and you plan to take
two of them in winter term. How many ways can these
courses be selected?
Another example
 How many ways can a committee of five people be selected
from a group of eight people?