Question: In a horse race, how many different finishes among the first 3 places are possible if 13 horses are…

In a horse race, how many different finishes among the first 3 places are possible if 13 horses are running? (Exclude ties)

There are a possible [ ] different finishes.

Solution

To find the number of different finishes among the first 3 places in a horse race with 13 horses, we can use the concept of permutations. We need to arrange 3 horses out of 13 and the order matters since the position is important. First, calculate the number of permutations for choosing 3 horses out of 13. The formula for permutations is given by: \[ P(n, r) = \frac{n!}{(n-r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Substitute 13 for \( n \) and 3 for \( r \). \[ P(13, 3) = \frac{13!}{(13-3)!} \] Calculate \( (13-3)! \). \[ (13-3)! = 10! \] Calculate \( \frac{13!}{10!} \). \[ P(13, 3) = \frac{13 \times 12 \times 11 \times 10!}{10!} \] Cancel the \( 10! \) in the numerator and the denominator. \[ P(13, 3) = 13 \times 12 \times 11 \] Perform the multiplication. \[ 13 \times 12 = 156 \] \[ 156 \times 11 = 1716 \] Therefore, there are 1716 different finishes possible among the first 3 places.