[Math] Find #committee of 8 from 3 freshmen, 4 sophomores, 4 juniors, and 5 seniors contain at least one of each class

discrete mathematics

The question:

A student council consists of three freshmen, four sophomores, four juniors, and five seniors. How many committees of eight members of the council contain at least one member from each class?

My Attempt:

let A1 be the set of committee w/o freshmen. N(A1) = 13C8
let A2 be the set of committee w/o sophomores. N(A2) = 12C8
let A3 be the set of committee w/o juniors. N(A3) = 12C8
let A4 be the set of committee w/o seniors. N(A4) = 11C8

Then, N(committee w/ at least one member of each group) = 
   Total # possible committee of 8 - N(A1 u A2 u A3 u A4)

So, from the Inclusion/Exclusion principle, 

N(A1 u A2 u A3 u A4) = N(A1) + N(A2) + N(A3) + N(4) 
                       - N(A1 n A2) - N(A1 n A3) - N(A1 n A4)
                       - N(A2 n A3) - N(A2 n A4) - N(A3 n A4)
                       + N(A1 n A2 n A3 n A4)

But, the question doesn't have enough information for me to find out:

N(A1 n A2) … N(A3 n A4) and N(A1 n A2 n A3 n A4)

Is there any other way to solve this question? Or to find the missing information above?

Best Answer

You can have a committee with

  • No freshmen
  • No sophomores
  • No juniors
  • No seniors
  • Neither freshmen nor sophomores
  • Neither freshmen nor juniors
  • Neither freshmen nor seniors
  • Neither sophomores nor juniors
  • At least one from each class

But as for all of the others, they're impossible with the composition of the council.

So the full answer would be:

$$N = {16 \choose 8} - {13 \choose 8} - 2{12 \choose 8} - {11 \choose 8} + 2{9 \choose 8} + 2{8 \choose 8}.$$

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