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Chapter 2: Problem 4

In how many ways can five red balls, four blue balls, and four white balls be placed in a row so that the balls at the ends of the row are the same color?

Short Answer

Expert verified
The total number of ways is \(\frac{11!}{3!4!4!} + \frac{11!}{5!2!4!} + \frac{11!}{5!4!2!}\).

Step by step solution

01

- Determine possible colors for the ends

The balls at the ends must be the same color. There are three color options: red, blue, and white.
02

- Calculate the number of arrangements with red ends

If the ends are red, fix two red balls at the ends and arrange the remaining 3 red, 4 blue, and 4 white balls. The total number of balls to arrange is 11. So, the calculation is based on the formula for permutations of multiset: \[\frac{11!}{3!4!4!}\].
03

- Calculate the number of arrangements with blue ends

If the ends are blue, fix two blue balls at the ends and arrange the remaining 5 red, 2 blue, and 4 white balls. The total number of arrangements is \[\frac{11!}{5!2!4!}\].
04

- Calculate the number of arrangements with white ends

If the ends are white, fix two white balls at the ends and arrange the remaining 5 red, 4 blue, and 2 white balls. The total number of arrangements is \[\frac{11!}{5!4!2!}\].
05

- Add up all possible arrangements

Summing all possibilities, the total number of ways to arrange the balls is \[\frac{11!}{3!4!4!} + \frac{11!}{5!2!4!} + \frac{11!}{5!4!2!}\]. Calculate this to find the total.

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Key Concepts

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

multiset permutations
Multiset permutations refer to the arrangements of a set of items where some items may be identical. Unlike distinct permutations where every item is unique, multiset permutations must account for repeated items. This often involves dividing out the factorial of the counts of each repeated item from the total factorial of the items. For example, in the context of arranging five red balls, four blue balls, and four white balls, the formula \(\frac{11!}{5!4!4!}\) is used. Here, 11 represents the total number of balls, and 5, 4, and 4 represent the counts of the red, blue, and white balls, respectively.
combinatorial counting
Combinatorial counting helps us determine the number of ways to arrange or select items. In problems like the one with colored balls, we use combinatorial counting to sum up all possible arrangements. We considered the cases where the ends are red, blue, or white separately. This ensures we don't double-count any arrangement and follow a systematic approach. By calculating the number of possible ways for each scenario and then adding them together, we get the total number of arrangements.
factorial calculations
Factorial calculations are crucial in combinatorial problems. The factorial of a number n, written as \(n!\), is the product of all positive integers up to n. Factorials help determine the number of ways to arrange n items. For instance, \(11!\) gives the total number of ways to arrange 11 items. When dealing with repeated items, we divide by the factorial of the counts of each repeated set to adjust for those repetitions, like in \(\frac{11!}{5!4!2!}\).
applied combinatorics
Applied combinatorics uses mathematical methods to solve real-world problems involving counting and arrangement. In the exercise, we applied combinatorial principles to find the number of ways to arrange colored balls under specific conditions. This involves breaking the problem down into manageable parts, applying formulas for permutations, and adding results for comprehensive solutions. The methods and formulas used here provide a robust framework for solving various arrangement problems in different contexts.

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Most popular questions from this chapter

A gas of \(N\) identical particles is free to move among \(M\) distinguishable lattice sites on a lattice with volume \(V\), such that each lattice site can have at \(N \ll M\) most one particle at any time. The density of lattice sites is \(\mu=M / V\). Assume that and that all configurations of the lattice have the same energy. (a) Compute the entropy of the gas. (b) Find the equation of state of the gas. (Note: the pressure of an ideal gas is an example of an entropic force.)

Fifteen boys go hiking. Five get lost, eight get sunburned, and six return home without problems. (a) What is the probability that a sunburned boy got lost? (b) What is the probability that a lost boy got sunburned?

Find the number of permutations of the letters in the word, MONOTONOUS. In how many ways are four O's together? In how many ways are (only) 3 O's together?

Various six digit numbers can be formed by permuting the digits 666655 . All arrangements are equally likely. Given that a number is even, what is the probability that two fives are together?

A lattice contains \(N\) normal lattice sites and \(N\) interstitial lattice sites. The lattice sites are all distinguishable. \(N\) identical atoms sit on the lattice, \(M\) on the interstitial sites, and \(N-M\) on the normal sites ( \(N \gg M>1\) ). If an atom occupies a normal site, it has energy \(E=0\). If an atom occupies an interstitial site, it has energy \(E=\varepsilon\). Compute the internal energy and heat capacity as a function of temperature for this lattice.
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