Find the number of ways in which the committee can be formed if chemist X,Physicist Y and Mathematician Z(i.e. all three) cannot work together. In a university there are 2 Chemists,3 Physicists and 10 Mathematicians.The President of University wants to form a scientific committee of 6 members among them.Find the number of ways in which the committee can be formed if chemist X,Physicist Y and Mathematician Z(i.e. all three) cannot work together. * * * If we remove X,Y,Z and select 6 members from rest 12 members,we get $\binom{12}{6}=924$ ways but the answer given is 4785 ways.I am confused.

The answer should be the total number of ways that a committee can be formed, minus the number of ways a committee could be formed that has all three of X, Y, and Z.

The number of ways a committee can be formed is clearly 15 choose 6. If X, Y, and Z are all in the committee, then the rest of the committee is determined by which three out of the 12 remaining are chosen, so the number of committees that have X, Y, and Z is 12 choose 3. So our answer is ${15 \choose 6} - {12 \choose 3} = 4785$.