Final answer:
An IRS auditor can select 4 of 9 tax returns for an audit in 126 ways, and there are 240,240 ways to form a jury consisting of 5 men and 7 women from a pool of 14 men and 10 women.
Step-by-step explanation:
The questions asked are related to the field of combinatorics, a branch of mathematics dealing with counting, both as a means and an end in obtaining results, and certain properties of finite structures. To determine how many ways an IRS auditor can select 4 of 9 tax returns for an audit, we use the combination formula since the order of selection does not matter:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of items to choose from, k is the number of items to choose, and '!' denotes factorial. Here, n is 9 and k is 4.
C(9, 4) = 9! / (4!(9-4)!) = 126
For the pool of possible jurors consisting of 14 men and 10 women, the question asks for the number of different juries consisting of 5 men and 7 women. For the men, we use C(14, 5) and for the women, we use C(10, 7), and we multiply the two results to find the total number of different juries:
C(14, 5) * C(10, 7) = (14! / (5!(14-5)!)) * (10! / (7!(10-7)!))= 240,240.