Combinations and Permutations of Cards

In summary, the conversation is about dividing a deck of 52 cards into piles of 3 and determining the number of possible combinations. It is solved by choosing a number for the first pile and then finding the number of ways to split up the remaining cards for each value of k. If 0 is not allowed, the procedure is the same, but the cases where any pile has 0 are omitted. The total number of combinations is calculated by adding up the number of ways for each value of k, which results in (50x51)/2.
  • #1
Baron
4
0
Hey guys,
I have a problem relating to combinations and permutations.In how many ways can I divide a deck of 52 cards into piles of 3 with each pile containing any number?
for example 50,1,1 or 45,6,1
Thanks in advance
 
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  • #2
Is 0 allowed for one of the piles? If you look at in two steps it is fairly straightforward.
Step 1, choose a number (k) for the first pile. There are now 52-k left for the other two piles. Assuming 0 is allowed, there are 52-k ways to split up these cards. Now just add them up for k ranging from 0 to 52. It will sum to 52x53/2.

If 0 is not allowed, the procedure is the same, just omit the cases where any pile has 0.
 
  • #3
Thanks,0 is not allowed
 
  • #4
Can you explain this."Now just add them up for k ranging from 0 to 52".I'm a little lost.(not a math expert)
 
  • #5
Since 0 is not allowed. Put 50 in the first pile, then there is exactly 1 way of sorting the other two piles. Put 49 in the first pile and there are 2 ways of sorting the other 2 piles, ..., put 1 in the first pile and there are 50 ways of sorting into the other two piles. Net result 1+2+3+...+50 = (50x51)/2.
 

Related to Combinations and Permutations of Cards

1. What is the difference between combinations and permutations?

Combinations and permutations are both ways to arrange objects in a specific order. The main difference is that combinations do not take into account the order of the objects, while permutations do. For example, if you have three cards (A, B, and C), the combination ABC would be the same as BAC, while the permutations would be different. ABC would be different from BAC, and so on.

2. How do you calculate the number of combinations of cards?

To calculate the number of combinations of cards, you can use the combination formula, which is nCr = n! / (r!(n-r)!), where n is the total number of cards and r is the number of cards you want to choose. For example, if you have 52 cards in a deck and want to choose 5 cards, the calculation would be 52C5 = 52! / (5!(52-5)!) = 2,598,960 combinations.

3. Can you have repetitions in combinations and permutations?

In combinations, repetitions are not allowed, meaning each object can only be chosen once. However, in permutations, repetitions are allowed, meaning an object can be chosen more than once in a specific arrangement. For example, in a combination, you cannot have AAA, but in a permutation, you can have AAA if there are three A cards in the deck.

4. How do you calculate the number of permutations of cards?

The formula to calculate the number of permutations of cards is nPr = n! / (n-r)!, where n is the total number of cards and r is the number of cards you want to arrange. For example, if you have 52 cards in a deck and want to arrange 5 cards, the calculation would be 52P5 = 52! / (52-5)! = 311,875,200 permutations.

5. Can combinations and permutations be used in other scenarios besides cards?

Yes, combinations and permutations can be used in many different scenarios, such as in probability, statistics, and computer science. They are used to calculate the number of possible outcomes in a specific situation and are often applied in fields that involve counting and arranging objects or events.

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