Answered: If I have 4 paintings and 3 boxes, how…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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If I have 4 paintings and 3 boxes, how many ways are there to distribute them so:

1. both the paintings and the boxes are indistinguishable

2. both the paintings and the boxes are distinguishable

3. the paintings are distinguishable and the boxes are indistinguishable

4. the paintings are indistinguishable and the boxes are distinguishable

Expert Solution

Step 1

As per honor code, we will only answer first three parts.

Total number of paintings = 4

Total number of boxes = 3

Number of ways to distribute them such that both the paintings and the boxes are indistinguishable:

$\begin{array}{rcl} N (Both are indistinguishable) & = & \frac{7!}{4! \times 3!} \\ = & \frac{7 \times 6 \times 5 \times 4!}{4! \times 3 \times 2} \\ = & 35 ways \end{array}$

Step 2

Total number of paintings = 4

Total number of boxes = 3

Number of ways to distribute them such that both the paintings and the boxes are distinguishable:

$\begin{array}{rcl} N (Both are distinguishable) & = & 7! \\ = & 5040 ways \end{array}$

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