probability that exactly two envelopes will contain a card with a matching color 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

Suppose both the cards and envelopes are rearranged (without changing the pairings) so that all red envelopes are on the left and all green envelopes are on the left. Now focusing on the cards on the left, we want the probability that exactly one of these is red (then by symmetry there will also be one matching pair on the right).
There are $\binom{5}{1}=5$ ways of choosing one red card out of five, and $\binom{5}{4}=5$ ways of choosing four green cards out of five. There are $\binom{10}{5}$ ways of choosing the five cards on the left from the set of ten.
Hence, the probability is
$$\mathbf{Pr}[2\text{ cards match envelopes}]=\frac{\binom{5}{1}\binom{5}{4}}{\binom{10}{5}}=\frac{5\times5}{252}=\frac{25}{252}$$
