how many committees are possible?

Question

A class consists of 4 seniors, 5 juniors, and 1 sophomore. 4 of these 10 students will be chosen to serve on a committee. If no more than 1 senior may serve on the committee, then how many committees are possible?

Answer 1

we choose 1 senior from the 4 seniors, total possibility is 4, and we need choose another 3 out of the rest of 6 students, that's 6x5x4/(3x2x1)=20, so the possibility is 4x20=80 if 1 senior is selected.

if no senior is selected, than choose all 4 out of 6, that's 6x5x4x3/(4x3x2x1)=15.

So the total possibility of no more than 1 senior is selected is 80+15=95.

Answer 2

Number of commities: If now any sinyour then N₁ = C_6,4 = 15. If one sinyour then number of commities is

N₂ = C_4,1C_6,3 = 4·20 = 80.

So, total number of possible comitties is N = 15 + 80 = 95