Four accounting majors, two economics majors, and three marketing majors have interviewed for five different managerial positions with a large company. Find the number of different ways that five of these people could be hired if the first two positions are to be filled by an accounting manager and an assistant accounting manager, the third position is to be filled by an economics manager, and the last two positions are to be filled by a marketing manager and an assistant marketing manager.
To solve this problem, we can break it down into smaller steps.
Step 1: Select an accounting manager and an assistant accounting manager from four accounting majors. Since the order doesn't matter in this selection (i.e., once someone is selected as an accounting manager, their role is fixed), we can use combinations. There are 4 accounting majors to choose from, and we want to choose 2, so the number of ways to select an accounting manager and an assistant accounting manager is C(4, 2) = 6 (where C(n, r) represents the number of combinations of n things taken r at a time).
Step 2: Select an economics manager from two economics majors. Since there is only one position and the order doesn't matter, we can use combinations again. There are 2 economics majors to choose from, so the number of ways to select an economics manager is C(2, 1) = 2.
Step 3: Select a marketing manager and an assistant marketing manager from three marketing majors. Again, we can use combinations since the order doesn't matter. There are 3 marketing majors to choose from, and we want to choose 2, so the number of ways to select a marketing manager and an assistant marketing manager is C(3, 2) = 3.
Step 4: Multiply the results from the previous steps to get the total number of ways to fill all the positions. Since each step represents independent selections, we can multiply the numbers together:
6 (accounting managers) x 6 (assistant accounting managers) x 2 (economics manager) x 3 (marketing managers) x 3 (assistant marketing managers) = 216.
Therefore, there are 216 different ways to hire five people for these five positions.