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You have two bags of $w_1$ and $w_2$ number of coins with face value from $1$ to $n$ and each coin occurring at most once in each bag (example $1$ can occur in both bags but not in same bag twice).

What is the probability that there are $w$ common coins?

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closed as off-topic by Graham Kemp, colormegone, Harish Chandra Rajpoot, user91500, Stefan Mesken Mar 16 '16 at 4:54

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  • $\begingroup$ Are we to assume each bag is a random sample of different coins, with each equally likely? In anything resembling real life, this is not likely: there are lots more (US or Canadian) 25 than 50 cent pieces in circulation. $\endgroup$ – Robert Israel Mar 16 '16 at 2:43
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The probability that there are w pairs of commons coins in the two bags would be:$${w_1\choose w}*{n-w_1\choose {w_2-w}}\over{n \choose w_2}$$

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Hint: Count the ways to select $w$ coins common to bag-1 and $w_2-w$ uncommon coins to fill the second bag, out of all the ways to select $w_2$ coins in the second bag.

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  • $\begingroup$ Ninja'd by browngreen $\endgroup$ – Graham Kemp Mar 16 '16 at 2:58

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