Question

There are 18 finalists in a singing competition. The top six singers receive prizes. How many ways can the singers finish first through sixth​?

Answer 1

In this case, the order matters, so this is a permutation problem.
You need to pick 6 out of 18.

$_{n}P_{r} = \dfrac{n!}{(n - r)!}$

$_{18}P_{6} = \dfrac{18!}{(18 - 6)!}$

$_{18}P_{6} = \dfrac{18!}{12!}$

$_{18}P_{6} = \dfrac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12!}{12!}$

$_{18}P_{6} = 18 \times 17 \times 16 \times 15 \times 14 \times 13$

$_{18}P_{6} = 13,366,080$