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# how many possible combinations are there for a deck of cards

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how many possible combinations are there for a deck of cards

Jan 3, 2015

#2
+10

Hi Tetration, you have given the number of permutations NOT combinations.

That is you have said:  Given 52 cards how many ways can they be ordered.

this is 52!

or it is also    52P52

The permutation calculation is available on any scietnific calculator.

In the web2 calc it can be entered as     nPr(52,52)

$${\left({\frac{{\mathtt{52}}{!}}{({\mathtt{52}}{\mathtt{\,-\,}}{\mathtt{52}}){!}}}\right)} = {\mathtt{80\,658\,175\,170\,943\,878\,571\,660\,636\,856\,403\,766\,975\,289\,505\,440\,883\,277\,824\,000\,000\,000\,000}}$$

NOW, in mathematics (probability) Combination has a very specific meaning

It means how many subsets of a specific size can be chosen from the original set.

Order is NOT important.

So in order to answer this I need to know how many cards you will be chosing from the original 52.

If you are choosing all 52 the answer is just 1 because there is only one combination of 52 cards  that can be chosen from 52 cards.  That is, you must choose ALL of them.

However, say you wanted to choose 7 cards.

Then the number of possible combinations would be  52C7

$${\left({\frac{{\mathtt{52}}{!}}{{\mathtt{7}}{!}{\mathtt{\,\times\,}}({\mathtt{52}}{\mathtt{\,-\,}}{\mathtt{7}}){!}}}\right)} = {\mathtt{133\,784\,560}}$$

.
Jan 4, 2015

#1
+5

If you have a usual French deck consisting of 52 different card, and want to know how many different ways to list all the cards, you can use factorial.

The total # of different ways to list all the 52 cards, such that you have a 1st card, 2nd card and so forth, is 52!

52! = 52*51*50*49* ... *4*3*2*1 ≈ 8,1 *1067.

Why is this?

Your 1st card have to be one of the 52 cards: 52 possibilites.

Your 2nd card have to be one of the remaining 51 cards: a total of 52*51 possibilities.

Your 3rd card have to be one of the remaining 50 cards: a total of 52*51*50 possibilities.

Your 4th card have to be one of the remaining 49 cards: a total of 52*51*50*49 possibilities.

And so on until there is a single card left.

Jan 3, 2015
#2
+10

Hi Tetration, you have given the number of permutations NOT combinations.

That is you have said:  Given 52 cards how many ways can they be ordered.

this is 52!

or it is also    52P52

The permutation calculation is available on any scietnific calculator.

In the web2 calc it can be entered as     nPr(52,52)

$${\left({\frac{{\mathtt{52}}{!}}{({\mathtt{52}}{\mathtt{\,-\,}}{\mathtt{52}}){!}}}\right)} = {\mathtt{80\,658\,175\,170\,943\,878\,571\,660\,636\,856\,403\,766\,975\,289\,505\,440\,883\,277\,824\,000\,000\,000\,000}}$$

NOW, in mathematics (probability) Combination has a very specific meaning

It means how many subsets of a specific size can be chosen from the original set.

Order is NOT important.

So in order to answer this I need to know how many cards you will be chosing from the original 52.

If you are choosing all 52 the answer is just 1 because there is only one combination of 52 cards  that can be chosen from 52 cards.  That is, you must choose ALL of them.

However, say you wanted to choose 7 cards.

Then the number of possible combinations would be  52C7

$${\left({\frac{{\mathtt{52}}{!}}{{\mathtt{7}}{!}{\mathtt{\,\times\,}}({\mathtt{52}}{\mathtt{\,-\,}}{\mathtt{7}}){!}}}\right)} = {\mathtt{133\,784\,560}}$$

Melody Jan 4, 2015
#3
0

Thanks Chris Jan 4, 2015