/*! 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 1 In how many ways can 10 (identic... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 1

In how many ways can 10 (identical) dimes be distributed among five children if (a) there are no restrictions? (b) each child gets at least one dime? (c) the oldest child gets at least two dimes?

Short Answer

Expert verified
The number of ways to distribute the dimes are: (a) \({{14}\choose{4}}\) ways, (b) \({{9}\choose{4}}\) ways, and (c) \({{12}\choose{4}}\) ways.

Step by step solution

01

No restrictions

Applying the stars and bars theorem with 10 (identical) dimes and 5 children, the answer would be \({{10 + 5 - 1}\choose{5 - 1}} = {{14}\choose{4}}\) where 14 represents the total 'stars and bars', 10 being the stars (dimes) and 4 being bars (children - 1) as bars are one less than the number of children.
02

Each child gets at least one dime

In this case, each of the five children initially gets 1 dime which leaves 5 dimes left to distribute without restrictions. Hence, the total number of combinations would be \({{5 + 5 - 1}\choose{5 - 1}} = {{9}\choose{4}}\). Here, 9 represents the total 'stars and bars', 5 being the stars (remaining dimes) and 4 being bars (children - 1).
03

The oldest child gets at least two dimes

In this scenario, two dimes are given to the oldest child which leaves 8 to be distributed among all the 5 children. Hence, the number of distributions is \({{8 + 5 - 1}\choose{5 - 1}} = {{12}\choose{4}}\). Here, 12 represents the total 'stars and bars', 8 being the stars (remaining dimes) and 4 being bars (children - 1).

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

a) How many distinct paths are there from \((-1,2,0)\) to \((1,3,7)\) in Euclidean three-space if each move is one of the following types? \((\mathrm{H}):(x, y, z) \rightarrow(x+1, y, z)\) \((\mathrm{V}):(x, y, z) \rightarrow(x, y+1, z)\) \((\mathrm{A}):(x, y, z) \rightarrow(x, y, z+1)\) b) How many such paths are there from \((1,0,5)\) to \((8,1,7) ?\) c) Generalize the results in parts (a) and (b).

How many ways are there to pick a five-person basketball team from 12 possible players? How many selections include the weakest and the strongest players?

How many ways are there to place 12 marbles of the same size in five distinct jars if (a) the marbles are all black? (b) each marble is a different color?

Patter's Pastry Parlor offers eight different kinds of pastry and six different kinds of muffins. In addition to bakery items one can purchase small, medium, or large containers of the following beverages: coffee (black, with cream, with sugar, or with cream and sugar), tea (plain, with cream, with sugar, with cream and sugar, with lemon, or with lemon and sugar), hot cocoa, and orange juice. When Carol comes to Patter's, in how many ways can she order a) one bakery item and one medium-sized beverage for herself? b) one bakery item and one container of coffee for herself and one muffin and one container of tea for her boss, Ms. Didio? c) one piece of pastry and one container of tea for herself, one muffin and a container of orange juice for Ms. Didio, and one bakery item and one container of coffee for each of her two assistants, Mr. Talbot and Mrs. Gillis?

Sixteen people are to be seated at two circular tables, one of which seats 10 while the other seats six. How many different seating arrangements are possible?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

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.