<s>[INST] The probability of being dealt a 2 pair 5 card hand... Given a standard playing card deck of 52 cards, what is the probability of being dealt a 2 pair 5 card hand consisting exactly of one pair of face cards and one pair of NOT face cards is?
 [/INST] First, I'll solve the problem assuming that it doesn't matter whether or not the fifth card matches either of the pairs.

There are $3 \cdot 4 = 12$ face cards in the deck, and $52 - 12 = 40$ non-face cards.
The number of choices for the positions of the pairs is
$$\binom{5}{2} \cdot \binom{3}{2}$$
The probability of getting the desired cards in a specific order is:
$$P_0 = \frac{12}{52} \cdot \frac{3}{51} \cdot \frac{40}{50} \cdot \frac{3}{49} \cdot \frac{48}{48}$$
$$P_0 = \frac{3}{13} \cdot \frac{1}{17} \cdot \frac{4}{5} \cdot \frac{3}{49}$$
$$P_0 = \frac{3 \cdot 2 \cdot 2 \cdot 3}{13 \cdot 17 \cdot 5 \cdot 7 \cdot 7}$$
$$P_0 = \frac{36}{54145}$$
The final probability is simply the product of this and the number of choices for positions of the pairs:
$$P = P_0 \cdot \binom{5}{2} \cdot \binom{3}{2} = \frac{216}{10829}$$

If it does matter that the fifth card does not match either of the pairs, then the last term in our calculation of $P_0$ becomes $44/48$ instead of $48/48$, which gives us a final answer instead of $$ P = \frac{198}{10829}$$
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