# Principle of counting problem

How many committees of 5 can be chosen from 10 people if each committee includes Alice and excludes Bob?

How many committees of 5 can be chosen from 12 people if each committee (i) includes Alice; (ii) excludes Bob; (iii) includes Alice and excludes Bob; (iv) includes at least one of either Alice or Bob

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1) To form a committee of $5$ people which includes Alice, we need to choose just $4$ more members. Since it excludes Bob, we only have $8$ out of the $10$ people to choose from (Alice and Bob have been accounted for). So this can be done in ${8}\choose{4}$ ways.

2)Inclusion into a committee is to be regarded as one less person to choose and exclusion from a committee should be thought as one less person to choose from. The various cases are just a reapplication of these ideas once or more.

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Great mod. I gave you an upvote before seeing this because you had it right. Hope this helps OP. I had tried to comment on mine, but the edits didn't take because of timing. – Ross Millikan Nov 20 '10 at 4:23

@Timothy Wagner: here is my suggestion. Thanks for not doing the hardest one. There is a wide range in how people respond to homework questions, so don't take mine as gospel-make up your own mind. But the differences between us are small. It is in how big a hint to give.

I bet this is homework. Please tag as such.

1)You have 10 people to choose from. Alice is required and Bob is excluded. So there are really only 8 to think about.

2)iii is the same (with 10->12) as the first. i and ii and not so different

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@Ross: Thanks for the tips. I have been on the site for just a week, so I am still coming to terms with it. The OP did not mention that it was a HW problem and I thought was just looking at strategies to tackle such problems. Should I modify my answer? – Timothy Wagner Nov 20 '10 at 4:04
@ Timothy: u cud edit ur post just 2 give hints instead of the complete answer (applicable only for hw pblms) – user17762 Nov 20 '10 at 4:08
We see many people looking for homework answers. Most do not admit it is HW. I wouldn't modify the answer as s/he has probably seen it, just a thought for the future. Welcome aboard. I've only been here a couple of months, we are all trying to figure this out. – Ross Millikan Nov 20 '10 at 4:12
@Ross Millikan, Timothy Wagner: the general consensus is that you should not tag something as homework based on your guess that it is homework. You can ask the poster, though some people find such questions irritation (as a quick search through Meta will reveal). – Arturo Magidin Nov 20 '10 at 4:14
@Arturo: Do you recommend answering posts not tagged as homework in full? – Timothy Wagner Nov 20 '10 at 4:20