A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?\\
(A) $\binom{10}{3} \binom{7}{4} 2^4$\\
(B) $\binom{10}{3} \binom{7}{4}$\\
(C) $\binom{10}{3} 2^7$\\
(D) $\binom{10}{3} \binom{14}{4}$