Here's the question:
A 5-card hand is selected from a standard deck of playing cares. (A standard deck has 13 cards from each of
4 suits{clubs, diamonds, hearts, and spades. The 13 cards have face value 2 through 10, jack, queen, king,
or ace. Each face value is \kind" of card. The jack, queen, and king are \face cards.") How many hands contain exactly one pair?
And here's the provided answer:
We can view this as a sequence of events. First, select the face value for the pair. There are 13 choices
for this. Next, select which two cards of that face value will make up the pair. Thus, we want to select
2 cards from the 4 of that face value at this step. There are C(4; 2) = 6 ways to do this. Third, we need
to select the other three cards. These other three cards cannot contain any pairs. There are 48 ways
to select the rest of these cards, since we have to exclude the cards with the face value of the pair. The
next card has 44 options, since we cannot select any of the two face values used thus far. The third of
these three cards has 40 options, since we cannot use any of the 3 face values already used. However,
we are over counting by all possible ways to arrange these three cards, so we must divide out by the
number of arrangements, namely 3! = 6. The final answer is:
(13*6*48*44*40)/6
= 1098240
I understand all of the steps except one: how are we over counting? P(3,3) = 6 which implies that the ordering of the cards in the hand matters, which makes no sense to me because it's still the same hand regardless of what order it's in. So how are we over counting, where is 3!=6 coming from if not P(3,3).
Thanks!
OSFTW
Best Answer
Order does not matter, as you say. However, we counted the last three cards as if order does matter. We correct for this by dividing by $3!$.