SOLUTION: Out of 11 biologists and 6 physicists, a committee consisting of 8 biologists and 2 physicists is to be formed. In how many ways can this be done if two particular biologist canno

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Question 1098474: Out of 11 biologists and 6 physicists, a committee consisting of 8 biologists and 2 physicists is to be formed. In how many ways can this be done if two particular biologist cannot be on the committee together?
Answer by Edwin McCravy(20054) About Me  (Show Source):
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Out of 11 biologists and 6 physicists, a committee consisting of 8 biologists and 2 physicists is to be formed. In how many ways can this be done if two particular biologists cannot be on the committee together? 
So there are 2 "restricted" biologists and 9 non-"restricted" biologists.

So the committee consists of either:

Case 1: 1 restricted biologist, 7 non-resticted biologists, and 2 physicists.  

Choose the 1 restricted biologist in 2C1 ways.  (2 choose 1)
Choose the 7 non-restricted biologists in 9C7 ways. (9 choose 7)
Choose the 2 physicists in 6C2 ways. (6 choose 2)

That's (2C1)(9C7)(6C2).

          OR

Case 2: 8 non-restricted biologists, and 2 physicists.  

Choose the 8 non-restricted biologists in 9C8 ways. (9 choose 8)
Choose the 2 physicists in 6C2 ways. (6 choose 2)

That's (9C8)(6C2).

Total for both cases:  (2C1)(9C7)(6C2) + (9C8)(6C2)

Edwin