/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 The game of bridge is played by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 25

The game of bridge is played by 4 players, each of whom is dealt 13 cards. How many bridge deals are possible?

Short Answer

Expert verified
There are a total of 1431 possible bridge deals. This was calculated using the concept of stars and bars and dividing the 52 cards into 4 groups of 13 cards for each player (\( \binom{54}{2} = 1431\)).

Step by step solution

01

Divide the cards among the first 3 players

First, we need to find out how many cards can be divided among the first 3 players. We can use the stars and bars method. We have 52 cards (stars) and we need to divide them into 3 groups, implying we need two bars to separate them. So, for example, we could have (13 cards for player 1 | 13 cards for player 2 | 26 cards for player 3). This problem can be represented as finding the number of ways of arranging 52 stars and 2 bars. The total number of objects is 52+2=54; we need to choose the positions for the 2 bars. There are \( \binom{54}{2}\) ways to choose the positions for the 2 bars among the 54 objects.
02

Find the remaining cards for the fourth player

After dividing the cards among the first 3 players, the remaining cards are for the fourth player. If the first three players have 13 cards each, the fourth player receives the remaining 13 cards. Thus, there is only 1 way to distribute the cards for the fourth player in this case. In general, this should be a natural consequence of distributing the cards among the first 3 players. Once those are allocated, the fourth player will have a predetermined number of cards remaining. There is no choice left for the fourth player.
03

Calculate the total number of bridge deals

Since there are \( \binom{54}{2}\) ways to divide the cards among the first 3 players, and only 1 way to distribute the remaining cards to the fourth player, the total number of bridge deals is the same as the total distribution for the first three players. Therefore, the number of possible bridge deals is \( \binom{54}{2} = \frac{54!}{2!(52!)} = \frac{54 \times 53}{2} = 27 \times 53 = 1431\). So, there are a total of 1431 possible bridge deals.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How many 5 -card poker hands are there?

Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place?

How many different 7-place license plates are possible when 3 of the entries are letters and 4 are digits? Assume that repetition of letters and numbers is allowed and that there is no restriction on where the letters or numbers can be placed.

A student is to answer 7 out of 10 questions in an examination. How many choices has she? How many if she must answer at least 3 of the first 5 questions?

A president, treasurer, and secretary, all different, are to be chosen from a club consisting of 10 people. How many different choices of officers are possible if (a) there are no restrictions; (b) \(A\) and \(B\) will not serve together; (c) \(C\) and \(D\) will serve together or not at all; (d) \(E\) must be an officer; (e) \(F\) will serve only if he is president?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.