In a race in which twelve automobiles are entered and there are no ties, in how many ways can the first three finishers come in? ways

To determine the number of ways the first three finishers can come in among twelve automobiles, we can use the concept of permutations. The number of permutations of selecting 3 finishers from 12 is given by \( P(n, k) = \frac{n!}{(n-k)!} \), where \( n \) is the total number of items (automobiles) and \( k \) is the number of selections (finishers). Here, \( n = 12 \) and \( k = 3 \): \[ P(12, 3) = \frac{12!}{(12-3)!} = \frac{12!}{9!} = 12 \times 11 \times 10 = 1320 \] So, there are 1320 different ways the first three finishers can come in.