Question

There are 12 technicians and 9 chemists working in a research laboratory. In how many
different ways can they form a 4-member safety committee if:
a) there are no restrictions?
b) the committee must have 2 technicians and 2 chemists?
c) the committee must have at least 2 technicians?
d) the committee cannot be all chemists or all technicians?
e) Mr. Smith (a technician) is not allowed to serve on the same committee as his wife,
Mrs. Smith (a chemist)

Answer 1

a) (12+9 choose 4), since order doesn't matter.
b) (12 choose 2)*(9 choose 2)
c) (12 choose 2)*(10+9 choose 2)

Do you think you can do part d and e?

Answer 2

Hint for part d. We know that there must be at least one tech and one chemist in the 4 member committee. So, let's immediately add one of each.

So, 12*9*? ... What should the ? mark be?

Answer 3

Ok, so part d is: 12*9*(11+8 choose 2). Why? Well, we know that at least one of the 12 techs are a member, and at least 1 of the 9 chems are a member. That's where 12*9 comes in. Now, we only have 11 techs and 8 chems left, and we can choose them however we like, so that's where the (11+8 choose 2) comes in.

Answer 4

Part e is a little trickier, but I'll start you off on an approach you can use.

Break it down into cases. We have three cases to consider.

Case 1: Mr. Smith, a tech, is a member. Thus, Mrs. Smith, a chem, cannot be.
Case 2: Mrs. Smith, a chem, is a member. Thus, Mr. Smith, a tech, cannot be.
Case 3: Neither Mr. Smith nor Mrs. Smith is a member.

Answer 5

I never saw this yesterday, thanks for the help!