Math question?

Question

From a standard deck of 52 cards, how many different five card hands are possible if the five card hand must consist of

i)only spades or only red face cards?
ii) four jacks
iii) exactly three kings

please help thank you!

Answer 1

For each of these 3 hands, just take it step by step and figure out how many ways there are to do it.

i) only spades or red face cards:
I'm assuming that a hand of all spades or all red face cards would qualify, but a hand consisting of some of each type is not acceptable. A different responder said the answer is 19 C 5, which assumes that a hand can have both spades and hearts.

Here's my solution:
How many ways can it be all spades? That's the same as: How many ways can you draw 5 spades from all 13 spades? This is sometimes called "13, choose 5." Well, for the first card you draw, there are 13 choices, for the second there are 12, then 11, 10, and 9. Multiply these numbers together to get the total number of possible sequences. This can also be expressed as 13! / 8!

But we weren't asked how many SEQUENCES of 5 spades. We were asked how many DIFFERENT HANDS of 5 spades can be formed. And each possible hand is counted multiple times in the number just calculated. For example, the hand A 2 3 4 5 appears as that sequence, but it also appears as A 2 3 5 4 and as A 2 4 3 5 and A 2 4 5 3, etc. It turns out that each DIFFERENT HAND appears as 5 x 4 x 3 x 2 x 1 = 120 different SEQUENCES. So the number calculated in the preceding paragraph has to be divided by 120.

And to that number we have to add the number of ways to make a hand out of red face cards. By the same reasoning, we can calculate that by substituting 6 for 13 in the preceding 2 paragraphs, since there are 6 red face cards (Jack, Queen, King of Hearts and Diamonds).

Calculate those two numbers and add them together.

(13 x 12 x 11 x 10 x 9) / (5 x 4 x 3 x 2 x 1) +
(6 x 5 x 4 x 3 x 2) / (5 x 4 x 3 x 2 x 1)

ii) Well there's one way to have four jacks, and 48 ways to have a fifth card in the hand. So the answer is 1 x 48 = 48.

iii) There are 4 ways to select 3 kings. (In other words, there are 4 ways to decide which king is excluded.) Then the other 2 cards have to be drawn from the 58 non-kings. There are 48 ways to draw one card, times 47 ways to draw a second card. Then we have to divide by 2, because we have calculated the number of SEQUENCES, and each pair of cards has been counted twice.
Result: 4 x 48 x 47 / 2.

Answer 2

i) 13 spades and 6 red face cards in a deck, so 19C5 = 11628
ii) The hand must contain all four jacks, so you're left with 48 choices for the 5th cards, so the answer is 48.
iii) You need to eliminate the 4 kings, so you have 48 cards to choose from for your final 2 cards. So, part of the solution is 48C2. You need 3 of the 4 kings, so that's 4C3. I can't remember if you're supposed to multiply those or add them... (either 48C2 + 4C3, or 48C2 * 4C3)

Answer 3

????? are all 3 of these separate hands?