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mr_coffee

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Another example from the book I'm going over and I'm not exactly sure how they got their answer:

The game of poker is played with an ordinary deck of cards. Various five-card holdings are given special names.

a. how many 5 card poker hands contain two pair?

Well I'm looking at the list, and I see two pair defined as this:

Two cards of one denomination, two cards of a second denomination, and a 5th card of a thrid denomination.

They said:

consider forming a hand with two paris as a four-step process:

Step 1: Choose the two denominations for the pairs.

Step 2: Choose 2 cards from the small denomination

Step 3: choose 2 card from the larger denomination

Step 4: choose one card from the remianing.

When they say denomination do they mean like:

A,2,3,4,5,6,7,8,9,10,J,Q,K

also why do they say choose 2 cards rfom the small and large denomination? couldn't u do it by getting like 2 kings, and 2 10's, and 1 other card like a 2?

These are all large denominations.

When i did this problem i thought it would be solved like this:

You have 52 cards to pick from, but you only want 2 denominations, so

Step 1: 52 choose 2

Step 2: You already used 2 cards so you only have 50 to choose from now, so 50 choose 2

Okay now you should have 4 cards, now you only need 1 more card, but you can't choose a card you picked in step 1 or 2, so u must take 52-4 = 48, 48 choose 2

(52 choose 2) * (50 choose 2) * (48 choose 1)

But they did it the following way:

total number of hands with two pairs:

Step 1: 13 choose 2

Step 2: 4 choose 2

step 3: 4 choose 2

step 4: 44 choose 1

i'm not sure how they got this...

Also the 44 choose 1, I am not sure how they got 44. Any explanation would be great!