A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of three of a kind (three of one denomination, one of another denomination, and one of a third).

1

To find the number of different poker hands of three of a kind, we can break down the process into several steps:

Step 1: Choosing the denomination for the three of a kind.
Since there are 13 denominations in a standard deck of 52 cards, we have 13 options to choose from. Let's say we choose denomination X.

Step 2: Choosing the denomination for the remaining two cards.
Once we have chosen denomination X, there are 12 remaining denominations to choose from for the remaining two cards. We need to choose one of them. Let's say we choose denomination Y.

Step 3: Choosing the cards for the three of a kind.
In a standard deck of 52 cards, each denomination has four cards (one in each suit). To form a three of a kind, we need to choose three cards of denomination X. Since there are four cards of denomination X, we can choose any combination of three cards out of the four. This can be represented as choosing 3 out of 4, which can be calculated using combinations.

Step 4: Choosing the card of denomination Y.
After we have chosen the three cards of denomination X, we need to choose one card of denomination Y. There are four cards of denomination Y in the deck, so we have four options to choose from.

Step 5: Calculating the number of different poker hands of three of a kind.
To calculate the number of different poker hands of three of a kind, we multiply the number of options from each step together. So the total number of different poker hands of three of a kind is:

Number of options for step 1 * Number of options for step 2 * Number of options for step 3 * Number of options for step 4

= 13 * 12 * (4 choose 3) * 4

Now let's calculate this:

= 13 * 12 * (4! / (3! * (4-3)!)) * 4
= 13 * 12 * (4) * 4
= 7488

Therefore, there are 7488 different poker hands of three of a kind.