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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 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

Compute Quantity A

Quantity A is number of distinct pairs of judges, which can be calculated using the combinations formula, given by \(C(n, k) = \frac{n!}{k!(n-k)!}\). Here we are selecting 2 judges out of 20. So, \(n = 20\) and \(k = 2\). Hence, Quantity A = \(C(20, 2) = \frac{20!}{2!(20-2)!} = 190\)
02

Compute Quantity B

Quantity B is number of possible rankings of dogs from first to third place, which can be calculated using the permutations formula, given by \(P(n, r) = \frac{n!}{(n-r)!}\). Here we are selecting and arranging 3 dogs out of 10. So, \(n = 10\) and \(r = 3\). Hence, Quantity B = \(P(10, 3) = \frac{10!}{(10-3)!} = 720\)
03

Compare Quantity A and B

Now that we have computed both quantities, we can clearly see that Quantity B is greater than Quantity A as 720 > 190.

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