More card fun, Straight Flush, full house, have the answers just don't get it

[mr_coffee]

Hello everyone. I am having some more card troubles.

This is suppose to be on how many ways can a certain hand be constructed.
b: A straight flush can be constructed by choosing a suit in 4 ways, and choosing the lowest denomination in
9 ways (from A, 2, 3, 4, 5, 6, 6, 8 or 9, but not 10, since 10 J Q K A is a royal flush, not a straight flush). So
there are 36 straight flushes.

It doesn't look like he even used combinations in this but just applied the muliplciation rule, but this confusees me more because your "choosing" out of a hand so why not use combinations?

I would have thought it would have worked like this:
Step 1: Choose the suit
Step 2: Choose the denominations in that suit
An example of a straight flush is this:
Q♠ J♠ 10♠ 9♠ 8♠
but in this problem he says, but not 10. but clearly it can be constructed if you use the 10 card and you still won't end up with a royal flush.
another example would be:
7♥ 6♥ 5♥ 4♥ 3♥

Okay so Step 1: There are 4 suits total, you only need to pick 1 suit, and stick with that suit. So 4 choose 1.

Step 2: You need to now pick 5 cards of that suit, so you only have 13 choices, so (13 choose 5)

I get: (13 choose 5) * (4 choose 1) = 5148

If i restrict the card chocie to only the cards he said, that would be
A, 2, 3, 4, 5, 6, 6, 8 or 9
So another version i would get:
9 choose 5 * 4 choose 1 = 504, still not close to his answer.

If you can tell me the steps he used that would help, rather tahn my version where I said:
Step 1: Choose the suit
Step 2: Choose the denominations in that suit


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d: A full house can be constructed by choosing a denomination and 3 cards of that denomination in 13x4 = 52
ways, and choosing another denomination and 2 cards of that denomination in 12 x (4 choose 2) = 72. so there are 52 x 72 = 3744 Full hosues

an example of a full house is the following:
3♣ 3♠ 3♦ 6♣ 6♥

So here is what I thought you would do:
Step 1: Choose a denomination
Step 2: Choose 3 cards of that denomination
Step 3: Choose another denomination
Step 4: choose 2 cards from that denomination

So..
Step 1: You have (52 choose 1), because you have 52 cards to choose from.
Step 2: Now that you already chose your denomination, you must select 3 cards of that type, but you already picked 1, so now you have 2 chocies left. soo (4 choose 2)
Step 3: Choose another denomination that wasn't the one you picked in step 1, so you have (51 choose 1) or would it be (52-3 choose 1), because you already picked 3 cards? so your left with 49 cards, so maye its (49 choose 1)
Step 4: You already picked 1 denomination, so you only need 1 more so you have 3 choose 1

 

[mattmns]

For the straight flush:

Step 1. You are right, there are just 4.

Step 2. Here the book is choosing the lowest denomination and associating with it the next 4. What I am trying to say is that if you pick 2 as your denomination, then you are choosing flushes that have a 2,3,4,5,6. If you choose 9 as your denomination, then you are choosing flushes that have a 9,10,J,Q,K. Since there are 9 possible denominations in this sense, there are 4x9 = 36 possible straight flushes.
For the full house:

You are not quite doing it correct, or I should say, the easiest way. You wrote the steps right but you did not follow through correctly.

Step 1. When you choose a denomination, you have 13 possible denominations, and you are picking one, hence you have 13 choose 1 = 13.

Step 2. Now you want to choose 3 cards (suits) of that denomination, hence you have 4 choose 3 = 4.

Step 3. Next pick the other denomination (for the 2 of a kind). You have already picked one denomination, so now you have 12 left, and you are choosing one, hence you have 12 choose 1. = 12.

Step 4. You want to pick the cards (suits) of that denomination, and you have 4 choose 2 ways.

Now if you use the multiplication principle you get what the book has.

 

[mr_coffee]

Awesome thanks again! makes a lot of sense once you say it. Maybe i can get one on my own hah.

Actually pretty soon we'll run out of hands so on the exam I shouldn't have a problem :)