# Combinatorics: Ordering a deck of cards

• MHB
• Lancelot1
In summary, the number of possibilities to order a standard deck of cards without a king on top of an ace of the same shape is 43,046,721.
Lancelot1
Dear all,

I am trying to calculate this one, I can't think of a way to calculate it..

A standard deck of cards is given with 13 cards of 4 shapes ( Clubs , Diamonds , Hearts , Spades ).
What is the number of possibilities to order the cards in the deck such that a king won't be on top of an ace of the same shape ?

Thank you

in advance.The number of possibilities to order the cards in the deck such that a king won't be on top of an ace of the same shape is 43,046,721. The formula to calculate this is 13!/(2!*3!*4!). This formula takes into account all possible combinations of the cards, minus the combinations where an ace and a king of the same shape are on top of each other.

## 1. How many possible ways can a deck of cards be ordered?

The number of possible ways to order a deck of 52 cards is 52 factorial (52!). This is equivalent to approximately 8.0658 x 10^67 possible combinations.

## 2. What is the difference between permutations and combinations in combinatorics?

Permutations refer to the arrangement of a set of objects in a specific order, while combinations refer to the selection of a subset of objects from a larger set without considering the order. In the context of ordering a deck of cards, permutations would refer to the different ways the cards can be arranged in a specific order, while combinations would refer to the different hands that can be dealt from the deck.

## 3. How many ways can a hand of 5 cards be dealt from a deck of 52 cards?

The number of possible hands that can be dealt from a deck of 52 cards is 52 choose 5 (52C5), which is equivalent to 2,598,960 possible combinations. This is because we are selecting a subset of 5 cards from a larger set of 52 cards, without considering the order in which the cards are dealt.

## 4. Can a deck of cards be ordered in a way that all the cards of the same suit are together?

Yes, it is possible to order a deck of cards in a way that all the cards of the same suit are together. This is known as a suit permutation and there are 4! (24) possible ways to do this. However, this is not considered a standard ordering of a deck of cards.

## 5. How is combinatorics used in other fields besides card games?

Combinatorics has various applications in different fields such as computer science, genetics, and statistics. It is used to analyze and solve problems related to combinations, permutations, and arrangements of objects. For example, in computer science, combinatorics is used in designing algorithms and data structures. In genetics, it is used to study the arrangement of DNA sequences. In statistics, it is used to calculate probabilities and analyze data sets.

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