Question 896089: In how many ways can a research team of 5 members be formed from a group of 10 scientists consisting of 3 chemists and 7 physicists if the team must include exactly 2 chemists?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! There are 3*2 = 6 ways to pick 2 chemists assuming order matters. Order doesn't matter though which means you have 6/2! = 6/2 = 3 ways to pick 2 chemists.
Put another way, there are 3 ways to NOT pick a chemist.
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You have 5 slots. Two of these slots must be chemists. So you really have 5-2 = 3 slots left.
You cannot have another chemist (that last third chemist) on the team because you must have *exactly* 2 chemists. No more and no less.
So you have 3 slots left and 7 people to choose from to fill those three slots.
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There are 7*6*5 = 42*5 = 210 ways to pick 3 people from a pool of 7
This is if order mattered.
However, order doesn't matter, so you really have 210/3! = 210/6 = 35 ways to pick 3 people from a pool of 7.
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To sum up what we have so far:
We have 3 ways to pick two chemists
We have 35 ways to pick three physicists
So there are 3*35 = 105 ways to pick a team of five members.
Again, order does not matter.
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