1
$\begingroup$

Suppose that we have a $52$-card deck. We are interested to find how many different combinations there could be if we divide this $52$-card deck in two parts, so that in each part there are $2$ aces.

What I am thinking is that we have $4$ aces, we can choose two of them in $\dfrac{4!}{2!}=12$ ways.

Also $52$ cards, we can divide in two parts in $\dfrac{52!}{2!}$ ways, so I have to multiply $12$ by $\dfrac{52!}{2!}$? It is a huge number, so am I missing something?

  • 0
    Note that sometimes it is acceptable to give an answer in terms of factorials and combinatorial numbers, so you might not actually have to perform the multiplication.2012-06-24
  • 0
    If you are splitting the deck between Alice and Bob, the answer is $6\times 2^{48}$. If you are just splitting, the answer is $3\times 2^{48}$.2012-06-24

3 Answers 3

2

There are 3 ways to split the aces: SH vs DC, SD vs HC, and SC vs HD (that's Spades, Hearts, Diamonds, Clubs).

Having split the aces, each of the other 48 cards has 2 possible destinations. So the answer would be $$3\times2^{48}$$

  • 0
    sorry one question,why is here three way?2012-06-24
  • 0
    aa i understood right about aces,but about 48 card,would not it be $48!/2!$?2012-06-24
  • 0
    Suppose the deck had only 5 cards. Would the part of the answer for the 1 non-ace be $1!/2!$? or would it be $2^1$?2012-06-24
  • 0
    right thanks very much2012-06-24
  • 1
    As @Gerry Myerson said it changes if you distinguish one hand from the other. So the correct answer is either 3 * 2^48 or 6*2^48. In the case of 5 cards, the two possible answers are either 3* 2^1 or 6* 2^12012-06-24
0

There are $\binom{4}{2}$ possibilities for picking 2 aces out of 4 and $\binom{48}{24}$ possibilities for picking 24 out of 48 non-aces. So the answer is $\binom{48}{24} \binom{4}{2}=193485622098600$.

  • 1
    But it didn't say the two parts had to have 26 cards each.2012-06-24
  • 0
    so is it possible to express answer like this?$48!/2!$*$4!/2!$?2012-06-24
0

Take out the 4 Aces. Hand #1 gets 2 Aces. Hand #2 gets 2 Aces.

The 4 Aces Spades, Clubs, Hearts , Diamonds (SCHD)

Hand #1 possiblities SC, SH, SD, CH, CD, HD for 6 possibilities

The remaining 48 cards each has a 50% chance of going to either hand. 2^48

The total possible hands = 6*2^48

  • 0
    That's if you distinguish Hand 1 from Hand 2. The question just said how many combinations if you divide the deck in two parts, so I interpret that to mean you don't distinguish the two hands. But it's a matter of interpretation, and only OP knows for sure.2012-06-24