Question

Consider a well-shuffled deck of 52 playing cards. What is the number of five-card hand with twpairs(two cards of one rank, two cards of a second rank, and one card from a third rank)?

Answer 1

hmm...i might be over thinking this one.

So first i want to know how many pairs can be made with 4 cards of the same type (like 4 kings, or 4 two's, etc).

The answer to that is:\[\left(\begin{matrix}4 \\ 2\end{matrix}\right)=\frac{4\cdot3}{1\cdot{2}}=6\] So given one type of card, i can make 6 different pairs.

Now, how many pairs can be made total, with the 13 different types of cards? Thats 13*6=78 pairs.

Now, to make a hand with two different pairs in it, im gonna choose 2 of those 78 pairs. How many ways can i pick two of those 78 pairs?\[\left(\begin{matrix}78 \\ 2\end{matrix}\right)=\frac{78\cdot 77}{1\cdot 2}=3003\]

Now i just have to worry about the last card. It can be anything, EXCEPT for the 4 cards that I already have in my hand, so 3003*(52-4)=3003*48= 144144. I think this is correct, anyone want to double check feel free.

Answer 2

The answer is wrong.

Answer 3

bleh =/ i knew i might be over complicating it. what is the correct answer?

Answer 4

123552. My answer sheet says 13C2* 4C2* 4C2*11C1*4C1

Answer 5

mmm I don't understand why

Answer 6

4C2 comes from my first statement for sure. 13C2 comes from the fact there are 13 types of cards, and you are going to pick two of them. AH, thats where i messed up.

Answer 7

why you have to multiply all that combinations?

Answer 8

because all of the combinations are used to build a single hand. Its like the problem:

A guy has 5 shirts, 4 pairs of socks, 3 pants, and 2 hats, how many outfits can he make?

Since picking one of each of those items makes one outfit, you multiply. Its called the Multiplication Principle.

Answer 9

ahh
i will think of it thanks!

Answer 10

I do not understand the 13C2 part and others.

Answer 11

So basically, the 13C2 comes from choosing 2 of the 13 types of cards.
The two 4C2's come from picking 2 of the 4 cards in that type.
The 11C1 comes from picking one of the 11 remaining types of cards
and the 4C1 comes from picking one of the four cards of that type.

This builds the hand you are looking for.

Answer 12

but why 4C2 comes twice? This is so hard to understand.

Answer 13

because you have 2 different types of cards you need pairs for. Lets say you pick ace's and 10's. There are 4 10 cards, so \[\left(\begin{matrix}4 \\ 2\end{matrix}\right)\]is the number of ways you can make pairs. Then you need another 4C2 for the aces. You will always need two of them.

Answer 14

Thanks.

Answer 15

for the 13C2, why is it 13C2? I am confused.