Question

A bridge hand consists of 13 cards dealt from the 52 -card deck. Bridge involves four players named North, East, South, and West. How many ways can the cards be dealt so that the game can be played?

Short answer

There are \( \frac{52!}{(13!)^4} \) ways to deal the cards for a bridge game.

Step-by-step solution

Understand the Situation

We need to calculate the different ways to distribute 52 cards among four players equally. Each player gets 13 cards.

Calculate Number of Hands

First, determine how many cards each player receives. This is straightforward as there are 13 cards per player.

Choose Cards for the First Player

Select 13 cards out of 52 for the first player. The number of ways to do this is given by the combination formula: \( \binom{52}{13} \).

Choose Cards for the Second Player

Next, choose 13 cards for the second player from the remaining 39 cards. The number of ways to do this is \( \binom{39}{13} \).

Choose Cards for the Third Player

Then, choose 13 cards for the third player from the remaining 26 cards. The number of ways to do this is \( \binom{26}{13} \).

Assign Cards to the Fourth Player

The fourth player receives the last 13 cards automatically, so there is only 1 way to do this, \( \binom{13}{13} = 1 \).

Calculate Total Arrangements

Multiply the results of each of the previous steps to find the total number of ways to distribute the cards: \( \binom{52}{13} \times \binom{39}{13} \times \binom{26}{13} \times \binom{13}{13} \).

Key concepts

Combination formula

When dealing with problems in combinatorics, the combination formula is a powerful tool. It helps us calculate the number of ways to select a specific number of items from a larger set without considering the order. This is crucial in card games where the order of card distribution doesn't matter, but the selection does.
The combination formula is mathematically represented as:- \[ \binom{n}{r} = \frac{n!}{r! (n-r)!} \]- Here, \( n \) is the total number of items, \( r \) is the number of items to choose, and \(!\) denotes a factorial, (which is the product of all positive integers up to that number).
For example, in a game of bridge, you use this formula to determine how many ways you can select 13 cards from a 52-card deck. This is written as \( \binom{52}{13} \).
Remember, calculations involve significant numbers. The factorial of 52 is a very large number, but by breaking it down as described in the formula, we simplify the computation.

Card distributions

Card distribution is a fascinating aspect of games like bridge. It involves carefully dividing a deck of cards among players, ensuring each gets an equal share. In bridge, where 52 cards are involved, each of the four players receives 13 cards.
The process is systematic. Firstly, you choose 13 cards from the deck for the first player. There are \( \binom{52}{13} \) ways to do this. After that, only 39 cards remain in the deck, and the process repeats for the next player with \( \binom{39}{13} \) ways. This counting continues until all players have their cards. The final calculation is straightforward, as the last 13 cards are simply given to the last player in one way, \( \binom{13}{13} = 1 \).
This structured approach ensures that all players have the same number of cards, highlighting the elegance of combinatorics in card games.

Probability theory

Probability theory provides a framework for understanding how likely an event is to occur. In card games, it often relates to determining the odds of drawing particular cards from a deck.
When distributing cards in bridge, probability helps us calculate the chance of any specific arrangement of cards. You multiply the combinations calculated for each player to get the total number of possible distributions. This approach gives insight into how vast the possibilities are in a standard deck distribution. - By using probability theory, players can assess the likelihood of certain hands, which is key in developing strategies during gameplay.
While the specific arrangement for a single deal is one in a combinatorially large number, by understanding these principles, players can appreciate and utilize statistics to make more informed decisions in the game.