Question

In a race between 18 people, how many ways can the top 5 finishers be arranged?

Answer 1

Answer:

Step-by-step explanation:

This is permutation, since order matters.  The formula for us is

₁₈P₅ = $\frac{18!}{(18-5)!}$ which simplifies to

₁₈P₅ = $\frac{18*17*16*15*14*13!}{13!}$

The factorial of 13 cancels out on the top and bottom leaving you with

₁₈P₅ = 18 × 17 × 16 × 15 × 14

which comes to 1,028,160 ways

Another way to look at it is:  the first 5 people of 18 finish and the others you don't care about.  Once the first place person is first, there are only 4 of the 18 left to finish in second place.  Then there are only 3 left to finish in third place, etc.  So if we use that reasoning, we don't even need to use the formula, we can just say

18 * 17 * 16 * 15 * 14 and those are the first 5 people of 18 to finish.