Problem Solving

Pursuit wrote:3 Freshmen
5 Sophomores
7 Juniors
9 Seniors

To form a committe with 8 members
At most 4 Sophomores

We have to select 8 members from (3 + 5 + 7 + 9) = 24 candidates
Hence, total number of possible selections = 24C8

But some of these selections will have all the 5 sophomores in them which is not permitted by the condition "at most 4 Sophomores". We need to discard those selections.
Number of such selections = (number of ways to select all the 5 sophomores)*(number of ways to select (8 - 5) = 3 members from (24 - 5) = 19 candidates) = 1*19C3 = 19C3

Hence, number os valid selections = 24C8 - 19C3