A hand of 5 cards is dealt from a standard deck of 52 cards. the number of ways

A poker hand consisting of 5 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 2 face cards.

What would be the probability?

First, we calculate the nubmer of ways the first two cards can be face cards, and the rest not: there are 12 face cards in the deck (4 Jacks, 4 Queens, and 4 Kings), and thus 12 possibilities for the first card and 11 possibilities for the second. The last three cards are dealt from the non-face cards, which have 40 possibilities for the first, 39 for the second, and 38 for the third. Thus there are 12*11*40*39*38 ways that exactly the first two cards are face cards. There are the same number of ways for exactly the first and third card to be face cards, and the first and fourth, and so on, so the total number of ways to deal exactly 2 face cards is 12*11*40*39*38 times the number of ways to select exactly two positions in your hand to be face cards.

How do we find this? There are 5 possiblities for the first position and 4 for the second position, but note that the order does not matter - i.e. having face cards in the first and third positions gives you the same set of hands as having the cards in the third and first postions. Thus, mutiplying 5*4 counts each combination twice, so we must divide this number by 2 to get the total number of combinations for hand positions to be face cards, which is then 10. Since this is a small number, I shall list them so that we can check our logic:

first and second
first and third
first and fourth
first and fifth
second and third
second and fourth
second and fifth
third and fourth
third and fifth
fourth and fifth

This is 10 combinations, as expected. Thus the total number of distinct hand you can be dealt that contain exactly two face cards is 10*12*11*40*39*38, or 78,249,600. The total number of hands you can be dealt period is 52*51*50*49*48, or 311,875,200. Thus the probability that you will be dealt exactly two face cards is 78,249,600 / 311,875,200, which is exactly 209/833, which is approximately 25.09%