Question

How many committees of six can be formed out of a group of twelve Republicans and ten Democrats if no more than four are Democrats?

Answer 1

First count the possible D/R splits:

4D, 2R
3D, 3R
2D, 4R
1D, 5R
0D, 6R

Then for each of these x D, y R, count how many ways you can pick x out of 10 times the number of ways you can pick y out of 12.

Answer 2

The number of committees is given by:
\[12C6+(10\times 12C5)+(10C2\times 12C4)+(10C3\times 12C3)+(10C4\times 12C2)\]

Answer 3

kropot72 is correct. Do you understand combinatorial notation?