Question

Suppose 40 cars start at a car race. In how many ways can the top 3 cars finish the​ race? The number of different top-three finishes possible for this race of 40 cars is nothing. ​(Use integers for any number in the​ expression.)

Answer 1

To calculate the top three finishes in a race of 40 cars, we use permutations. The formula is 40P3 which gives us 40! / 37! or 40 x 39 x 38, resulting in 59,280 different possible top-three finishes.

Step-by-step explanation:

To determine in how many ways the top 3 cars can finish the race out of 40, we must calculate the number of permutations of 40 items taken 3 at a time. This is because the order in which the cars finish matters, making it a permutation problem, rather than a combination, where order would not matter.

The formula for permutations is given by nPr = n! / (n-r)! where n is the total number of items, and r is the number of items to arrange. In our case, n is 40, and r is 3, so we are looking for 40P3.

Calculating this, we get 40! / (40-3)! = 40! / 37! = 40 x 39 x 38, since all the terms in the factorial of 37 cancel out with the terms in the factorial of 40 till we're left with three terms. So, the number of different top-three finishes possible for this race of 40 cars is 40 x 39 x 38, which is 59280.