Question

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

• A. 18
• B. 16
• C. 12
• D. 15

Answer 1

The Correct Option is D

To solve this problem, we need to find the number of ways to distribute 8 identical books into 4 identical shelves where some shelves can be left empty. This is a combinatorial problem related to the partitioning of numbers. 

The number of ways of distributing \(n\) identical items into \(r\) identical groups is equivalent to finding the number of partitions of \(n\) into at most \(r\) parts, each of which is a non-negative integer.

In our case, \(n = 8\) and \(r = 4\). We need to find the partitions of 8 into at most 4 parts.

The possible partitions are as follows:

• \(8 = 8\)
• \(8 = 7 + 1\)
• \(8 = 6 + 2\)
• \(8 = 6 + 1 + 1\)
• \(8 = 5 + 3\)
• \(8 = 5 + 2 + 1\)
• \(8 = 5 + 1 + 1 + 1\)
• \(8 = 4 + 4\)
• \(8 = 4 + 3 + 1\)
• \(8 = 4 + 2 + 2\)
• \(8 = 4 + 2 + 1 + 1\)
• \(8 = 4 + 1 + 1 + 1 + 1\)
• \(8 = 3 + 3 + 2\)
• \(8 = 3 + 3 + 1 + 1\)
• \(8 = 3 + 2 + 2 + 1\)

There are 15 different partitions, and hence the number of ways to arrange the 8 identical books into 4 identical shelves is 15.

Therefore, the correct answer is 15.

Answer 2

Solution: We need to count the ways of arranging 8 identical books into 4 identical shelves, allowing any number of shelves to be empty.

Step 1. 3 Shelves Empty: Only one shelf holds all 8 books:  
\((8, 0, 0, 0) \implies 1 \text{ way}\)

Step 2. 2 Shelves Empty: Two shelves hold the books in pairs of configurations:  
 \((7, 1, 0, 0), (6, 2, 0, 0), (5, 3, 0, 0), (4, 4, 0, 0) \implies 4 \text{ ways}\)

Step 3. 1 Shelf Empty: Three shelves hold the books in possible configurations:  
\((6, 1, 1, 0), (5, 2, 1, 0), (4, 3, 1, 0), (4, 2, 2, 0), (3, 3, 2, 0) \implies 5 \text{ ways}\)

Step 4. 0 Shelves Empty: All four shelves hold the books in possible configurations:  
\((5, 1, 1, 1), (4, 2, 1, 1), (3, 3, 1, 1), (3, 2, 2, 1), (2, 2, 2, 2) \implies 5 \text{ ways}\)

Adding up all the ways:
\(\text{Total} = 1 + 4 + 5 + 5 = 15 \text{ ways}\)

The Correct Answer is: 15 ways