A math department is allowed to tenure 4 of 17 eligible instructors. In how many ways can the selection for tenure be made?
Please show more than one approach.
Would I put 17 nCr 4 in the calculator?
Sure, why not?
To find the number of ways the selection for tenure can be made, we can apply the concept of combinations.
Approach 1: Using the Combination Formula
The combination formula is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of items to choose from, and r is the number of items to be chosen.
In this scenario, we have 17 eligible instructors, and we need to choose 4 for tenure. Using the combination formula, we can calculate the number of ways to make the selection:
C(17, 4) = 17! / (4!(17-4)!) = 17! / (4! * 13!) = (17 * 16 * 15 * 14) / (4 * 3 * 2 * 1) = 14,190
Therefore, there are 14,190 different ways to make the selection for tenure.
Approach 2: Using Multiplication Principle
Another approach is to use the multiplication principle.
We need to make 4 selections out of 17 eligible instructors.
For the first selection, we have 17 choices. For the second selection, we have 16 choices (one less instructor since we've already made one selection). For the third selection, we have 15 choices, and for the fourth selection, we have 14 choices.
Using the multiplication principle, we multiply these choices together:
17 * 16 * 15 * 14 = 43,680
Therefore, using this approach, there are 43,680 different ways to make the selection for tenure.
Thus, both approaches yield different numbers of ways to make the selection for tenure: 14,190 and 43,680, respectively.