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 Two similar questions from Gerald: I have a group of 7 men and 4 women. How many groups can be formed in which I would have: a) 3 men and 2 women and b) 5 people in which at least 3 among them would be men. I have a bag containing 7 black marbles and 5 white marbles. How many combinations of 5 marbles of which 3 are black and two are white are possible?

Hi Gerald,

I am going to look at a similar problem.

I have 8 tulip bulbs and 5 daffodil bulbs.

1. How many ways can I select 5 bulbs if I want 3 tulips and 2 daffodils?
2. How any ways can I select 6 bulbs if I want at least 4 to be tulips?

The key here is to know that the number of ways of choosing k things from n things is n!/[(n-k)! k!].

Thus the number of ways of choosing 3 tulip bulbs from 8 tulip bulbs is 8!/[(8-3)! 3!] = [8 * 7 * 6]/[3 * 2 * 1] = 56.

Likewise the number of ways of choosing 2 daffodil bulbs from 5 daffodils is 5!/[(5-2)! 2!] = [5 * 4]/[2 * 1] = 10.

Thus the number of ways of choosing 3 tulips and 2 daffodils is 5610 = 560.

For part b) first find the number of ways of choosing 4 tulip bulbs from the 8 tulip bulbs and then the number of ways of choosing 2 bulbs from the remaining 9 bulbs.

I hope this helps,
Penny

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