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

Answer: (D)

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