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Chapter 4: Problem 128

A veterinarian assigned to a racetrack has received a tip that one or more of the 12 horses in the third race have been doped. She has time to test only 3 horses. How many ways are there to randomly select 3 horses from these 12 horses? How many permutations are possible?

Short Answer

Expert verified
There are 220 ways to randomly select 3 horses out of 12 and 1320 possible permutations of the selected horses.

Step by step solution

01

Calculate combinations

Defining the symbols, n stands for the total number of horses, which is 12 and r is the number of horses that can be tested, which is 3. So the number of combinations of 3 horses that can be selected out of 12 is computed using the formula for combinations \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Thus, substituting n = 12 and r = 3, the number of combinations is \( \binom{12}{3} = \frac{12!}{3!(12-3)!}\) which gives 220 combinations.
02

Calculate permutations

Next, compute the number of ways these 3 horses can be arranged or the number of permutations. The formula for permutations \(P(n, r) = \frac{n!}{(n - r)!}\) is used for this. Here, n = 12 and r = 3, so the number of permutations is \(P(12, 3) = \frac{12!}{(12-3)!}\) which yields 1320 permutations.

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