I was trying to count how many cards with 4 of a kind I could get from a standard deck of 52 cards in a hand of length 5.
i.e. 4 of the same value (rank) and a card with a different value.
I was told the answer was $13 \times 12 \times 4 = 624$ but I was a little confused why.
The explanation given to me was because a four of a kind can be completely specified by a sequence saying
- The rank of the four cards
- The rank of the extra card
- The suit of the extra card
However, there is a specific reason I am confused. For me this reasoning seems to be undercounting. Why do I think that, well, I believe that the suit of the cards with same rank should be important too because a 4 of a kind hand with {8 spades, 8 spades, 8 spades, 8 spades, Q hearts} is a different 4 of a kind than {8 clubs, 8 clubs, 8 clubs, 8 clubs, Q hearts}
i.e. does the suit of the cards with the same kind not matter?
Why is it not $13 \times 4 \times 12 \times 4$ i.e.
- The rank of the four cards
- The suit of the four cards
- The rank of the extra card
- The suit of the extra card
I just don't understand why that is wrong.
Actually, that is wrong too, why is it not:
- The rank of the four cards
- The suit of each of the four cards in a combination way
- The rank of the extra card
- The suit of the extra card
As in why is the answer not:
$$ 13 \times C^4_1 \times C^4_1 \times C^4_1 \times C^4_1 \times 12 \times 4$$
or something that takes into account the different suits for the 4 of a kind cards?