I took this from challenge 25, question 3.
4 professors and 6 students are being considered for membership on a supervisory committee which must consist of 3 people. In how many ways can this committe be formed if it has to include at least one professor?
I post the answer immediately since it is anyway available on the challenge site.
The best way to approach this problem is to consider an unconstrained version of the question first: how many committees of 3 are possible? The answer is 10C3 = 10!/(7!*3!) = 120. From this figure we have to subtract the number of committees that consist entirely of students i.e. 6C3 = 6!/(3!*3!) = 20. The final answer is 10C3 - 6C3 = 120 - 20 = 100.
My question is this....Why can't I do this...
I need one professor in the team. So 4C1=4.
Then I need 2 guys from the remaining 3 professors and 6 students. Therefore i use 9C2=36.
Total nos of combinations= 36*4=144? Why is this wrong?
The correct way to do this prombem is
3 member can be 3 professors or (2 professors, 1 student) or (1 professor, 2 student) or 3 student
=4C3 + 4C2*6C1+ 4C1*6C2 + 6C3
= 4 + 6*6 +4*15+20