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Chapter 5: Problem 43

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 select 3 out of 12 horses, and for any set of 3 horses selected, there are 6 possible arrangements (permutations).

Step by step solution

01

Calculation of Combinations

To calculate the number of combinations of 12 horses taken 3 at a time, use the formula for combinations:\[C(n,r) = \frac{n!}{r!(n-r)!}\]\nHere, 'n!' is the factorial of n (product of all positive integers from 1 to n), 'r' is the number of items chosen at a time (3 in our case), and 'n' is the total number of items (12 in our case). Substituting the values in the formula, we get:\[C(12,3) = \frac{12!}{3!(12-3)!}\]
02

Calculation of Permutations

The number of ways to arrange the selected 3 horses is calculated using the formula for permutations (P), which is:\[P(n) = n!\]\nHere, n is the number of items to arrange. So, the number of permutations of 3 horses is:\[P(3) = 3!\]
03

Final Calculations

On calculating the factorials and simplifying, you get:\nCombination C(12,3) = 220 \nPermutation P(3) = 6

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