## Precalculus (6th Edition) Blitzer

The formula for $_{n}{{C}_{r}}$ has the same numerator as the formula for $_{n}{{P}_{r}}$ but contains an extra factor of $r!$ in the denominator.
We know that permutations of $n$ things taken $r$ at a time can be defined by the formula: ${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$ And combinations of $n$ things taken $r$ at a time can be defined by the formula: ${}_{n}{{C}_{r}}=\frac{n!}{r!\left( n-r \right)!}$ And by comparing both formulas it can be seen that r! is an extra factor. Therefore, there are r! times as many permutations of n things taken r at a time as combinations of n things taken r at a time.