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There are 9 pairs of socks. We choose 5 socks at random. what is the probability of getting at least 1 pair?

I computed the probability of the complement and came up with

$$ 1 - \frac{18\cdot 16\cdot 14\cdot 12\cdot 10}{18\cdot 17\cdot 16\cdot 15\cdot 14} $$

Is this correct?

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How many ways can you choose $5$ socks? $\binom {18}5$

How many ways there is no pair? $\binom952^5$ (choose 5 pairs, then choose a sock from each pair).

Then, the probability of getting no pairs is $$\frac{(9\cdot2)(8\cdot 2)(7\cdot 2)(6\cdot2)(10\cdot2)}{18\cdot17\cdot16\cdot 15\cdot 14}$$

This is exactly what you have computed. But we don't know how you got it.

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  • $\begingroup$ then choose a sock from one of the pairs $\endgroup$ – chouaib Oct 21 '14 at 14:16

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