For # 7-10, find the number of possible 4-card hands that contain the cards specified

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Answers:

Note: The answer to problem 7 is a single value. The comma is there to make the number more readable.

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Explanation for problem 7)

There are 26 red cards (13 diamonds + 13 hearts).

We have 26 ways to pick the first red card, and then 25 ways to pick the second red card. If order mattered, then we'd have 26*25 = 650 ways to do this. However, order doesn't matter. All that matters is the hand itself rather than the individual cards. By "hand" I mean the collection of cards, and not the literal physical hand holding them.

Since the count 650 is a double count, this means 650/2 = 325 is the correct count where order doesn't matter. The black cards will follow identical logic to get the same value. There are 325 ways to pick the two black cards. This is because there are an equal number of red and black cards, and we're selecting an equal number of both colors.

So we have 325*325 =  105,625 different hands possible.

To help show some context, there are 52C4 = 270,725 different ways to pick four cards without any restrictions. I'm using the nCr combination formula.

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Explanation for problem 8)

The face cards are Jack, Queen, King. There are 3 face cards per suit and 4 suits total, so 3*4 = 12 face cards in all.

We have 12*11*10 = 1320 permutations and 1320/(3!) = 1320/6 = 220 combinations. We side with combinations because like before, order doesn't matter. There are 220 different ways to pick the three face cards. Then we have 52-12 = 40 ways to pick the fourth non-face card.

Overall, we have 220*40 = 8800 different ways to pick exactly three face cards.

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Explanation for problem 9)

There are 13 diamond cards, so n = 13. We're filling r = 4 slots.

Use the nCr combination formula to find that 13C4 = 715

See the attached image below for more detailed steps.

We have 715 ways to pick all diamonds.

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Explanation for problem 10)

We'll build from problem 9. We found there are 715 ways to pick four diamonds. This is the same number of ways to pick four hearts, or four clubs, or four spades. The actual suit doesn't matter. So we have 4*715 = 2860 different ways to pick 4 cards of the same suit

In poker, having all cards of the same suit is known as a flush. Though with poker, it involves 5 cards instead of 4.