A group of 9 people is going to be formed into committes of
4, 3, and 2 people. how many committees can be formed if

a) a person can serve on any number of committees

We can pick people to serve on the 4-person committee any of 9C4 ways

For every one of those 9C4 ways,

we can pick people to serve on the 3-person committee any of 9C3 ways

That's 9C4*9C3 ways to pick people for the 4 and 3-person committees.

For every one of those 9C4*9C3 ways,

we can pick people to serve on the 2-person committee any of 9C2 ways

That's 9C4*9C3*9C2 ways to pick people for all the committees.

 9C4*9C3*9C2 = 126*84*36 = 381024 ways in all.

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b) no person can serve on more than one committee

We can pick people to serve on the 4-person committee any of 9C4 ways

For every one of those 9C4 ways,

we can pick people to serve on the 3-person committee any of 5C3 ways

That's 9C4*5C3 ways to pick people for the 4 and 3-person committees.

For every one of those 9C4*5C3 ways,

we can pick people to serve on the 2-person committee any of 2C2 ways

That's 9C4*5C3*2C2 ways to pick people for all the committees.

 9C4*5C3*2C2 = 126*10*1 = 1260 ways in all.

Edwin