06/04/2020 (2011: National Sprint Round, Problem 11)

Q: Ms. Marks is selecting students for a committee that must consist of three seniors and five juniors. Six senior volunteers are able to serve on the committee. What is the least number of junior volunteers needed if Ms. Marks wanted at least 100 different possible ways to pick the committee?

A: 6

The total number of possible ways to pick the committee is (Number of Senior Volunteers)C(3) ⋅ (Number of Junior Volunteers)C(5), where "C" is "Combinations." 

We know that there are 6 senior volunteers, and that we want to have at least 100 different possible ways to pick the committee, so we set up an equation: 6C3 ⋅ (Number of Junior Volunteers)C(5) ≥ 100

That simplifies to: 20 ⋅ (Number of Junior Volunteers)C(5) ≥ 100, which simplifies further to: (Number of Junior Volunteers)C(5) ≥ 5

We start plugging in numbers for the "Number of Junior Volunteers," starting with 5 (because you can't pick 5 juniors from less than 5 juniors:

5C5 = 1, which isn't ≥ 5

6C5 = 6, which IS ≥ 5

That means the answer is 6 junior volunteers