Permutations and Combinations

factorials

• $n=n(n-1)(n-2)\dots 1$

Permutations

• ${}^n{p}_r=\frac{n!}{(n-r)!}$

Restrictions

Combinations

• order don’t matter
• ${}^n{C}_r=\frac{n!}{r!(n-r)!}$

Identical Elements

• if we order n distinct elements, they can be arranged in n! ways

• if there are elements that are identical, we divide to account for duplicates

• “XXXABCDE”

  • can be rearranged $\frac{8!}{3!}$ times

In a circle

• if 8 people sit in a line: 8! arrangements
• if 8 people sit in a circle: $\frac{8!}{8}=7!$ arrangements