Suppose that 10 cards, of which 5 are red and 5 are green, are placed at random in 10 envelopes, of which 5 are red and 5 are green. Determine the probability that exactly two envelopes will contain a card with a matching color.
clearly $\Omega= 10!$
if I have $ 5c1$ ways red and $5c1$ ways green, How do I get other permutations? some hints please
Best Answer
HINT: Note that if $2$ red cards are in red envelopes, there are only $3$ red envelopes left to contain green cards, so at least $2$ green cards must end up in green envelopes. Similarly, if $2$ green cards are in green envelopes, then at least $2$ red cards must be in red envelopes. Thus, the only way to get exactly $2$ cards in matching envelopes is to have the red envelopes contain $1$ red and $4$ green cards, and the green envelopes contain $1$ green and $4$ red cards.
There are $5$ ways to choose which red card is to go in a red envelope, and then there are $5$ ways to choose a red envelope to contain it.
How many ways are there to pick $4$ green cards and distribute them amongst the remaining red envelopes?
Then how many ways are there to place the one remaining green card in a green envelope?
And finally, how many ways are there to distribute the remaining $4$ red cards amongst the remaining $4$ green envelopes?