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Chapter 1: Problem 17

Beim Skatspiel erh盲lt jeder der drei Spieler 10 Karten, w盲hrend die restlichen beiden Karten in den Skat gelegt werden. Auf wieviel verschiedene Arten k枚nnen die 32 Karten verteilt werden?

Short Answer

Expert verified
The total number of ways to distribute the 32 cards is \[ \binom{32}{10} \times \binom{22}{10} \times \binom{12}{10} \]

Step by step solution

01

- Identify Total Number of Cards

Determine the total number of cards being dealt and placed in the Skat. There are 32 cards in total.
02

- Distribute Cards Among Players

Each of the three players receives 10 cards. This means a total of 30 cards are distributed among the players, leaving 2 cards for the Skat.
03

- Calculate Number of Ways to Choose Cards for One Player

Calculate the number of ways to choose 10 cards out of 32 for the first player. This is given by the binomial coefficient: \(\binom{32}{10}\).
04

- Calculate Number of Ways to Choose Cards for Second Player

After choosing 10 cards for the first player, there are 22 cards left. Calculate the number of ways to choose 10 out of these 22 cards for the second player: \(\binom{22}{10}\).
05

- Calculate Number of Ways to Choose Cards for Third Player

After choosing cards for the first two players, there are 12 cards left. Calculate the number of ways to choose 10 out of these 12 cards for the third player: \(\binom{12}{10}\).
06

- Calculate Number of Ways to Choose Cards for Skat

Finally, the last 2 cards automatically go to the Skat. The number of ways to choose 2 out of 2 cards is 1: \(\binom{2}{2} = 1\).
07

- Calculate Total Number of Distributions

Multiply the results of the binomial coefficients from the previous steps to get the total number of ways to distribute the cards: \[ \binom{32}{10} \times \binom{22}{10} \times \binom{12}{10} \times 1 \]
08

- Simplify the Expression

Simplify the expression to find the final answer for the number of different ways to distribute the 32 cards.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Coefficient
Binomial coefficients are a key part of combinatorics. They show how many ways you can choose a specific number of items from a larger set. In general, for two numbers and , the binomial coefficient is written as \(\binom{n}{k}\) and is read as \

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