Counting: arrangements and selections

Factorials, permutations, combinations and arranging words with repeated letters. About 10 minutes, A-level.

What you will do

1. Counting without listing. Counting every possibility by hand explodes quickly. 5 people can queue in 120 different orders. Counting rules let us find the number without listing them.
2. Factorials. n! means n × (n-1) × ×s × 1: the number of ways to put n different things in order.
3. Order matters: permutations. Choose r things from n and arrange them: ^nP_r = n!(n-r)!.
4. Order does not matter: combinations. Choose r things from n with no arrangement: nr = n!r!(n-r)!. Divide the permutations by the r! ways to order each choice.
5. Words with repeated letters. Identical letters can be swapped without changing the word, so divide by k! for each letter that repeats k times.