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Chapter 14: Problem 5

At a dog show, there are 20 judges and 10 dogs in the final round. Quantity \(\mathbf{A}\) The number of distinct pairs of judges Quantity \(\mathbf{B}\) The number of possible rankings of dogs from first to third place a. Quantity A is greater. b. Quantity \(\mathrm{B}\) is greater. c. The two quantities are equal d. The relationship cannot be determined from the information given.

Short Answer

Expert verified
b. Quantity B is greater.

Step by step solution

01

Calculate Quantity A

Quantity A involves finding out the number of distinct pairs of 20 judges. As the order of the judges in a pair doesn't matter, we use combinations which is given by the formula \(\frac{n!}{k!(n-k)!}\). Here, \(n=20\) (total number of judges) and \(k=2\) (number of judges in each pair). So, Quantity A yields \(\frac{20!}{2!(20-2)!} = 190\).
02

Calculate Quantity B

Quantity B involves finding out the number of possible rankings of 3 dogs out of 10. In this scenario, order matters, so we use permutations. The formula for permutations is given by \(\frac{n!}{(n-k)!}\). Here, \(n=10\) (total number of dogs) and \(k=3\) (number of dogs being ranked). So, Quantity B yields \(\frac{10!}{(10-3)!} = 720\).
03

Compare Quantities

By comparing the two quantities: Quantity A is 190 and Quantity B is 720. Therefore, Quantity B is greater than Quantity A.

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