October 13, 2009 Combinations and Permutations

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) = n r (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) = 10 3 = 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=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=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?