# Help with simple counting argument

Hi (this is not homework! I just find it easy to make mistakes on these and need to know for a simple estimate at work.)

Out of a set S of 10,000 items, 50 of them are special. I need to know how many ways I can select a set of 50 items from S that contains at least 2 special items.

I calculate this as: "50 choose 2" x "9,998 choose 48".

Is this correct? Thank you! E

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## 1 Answer

No, you're overcounting the choices where more than $2$ of the chosen items are special, since you can designate different pairs of them as the "$2$ in $50$". I think the simplest way to calculate this is as the number of ways to choose $50$ from $10,000$ minus the number of ways to choose $50$ from $10,000$ with $0$ or $1$ special:

$$\binom{10000}{50}-\binom{9950}{50}\binom{50}{0}-\binom{9950}{49}\binom{50}{1}\;.$$

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Thank you very much for explaining - I understand! –  Erik Retson May 4 '11 at 16:15
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