Number of ways to arrange 4 novels and 2 biographies on a shelf chosen from 5 novels and 4 biographies

Introduction / Context:
This problem mixes combinations and permutations. You must first choose which books are part of the arrangement and then consider in how many ways they can be placed in order on the shelf. This is a common pattern in counting problems.

Given Data / Assumptions:

• There are 5 distinct novels.
• There are 4 distinct biographies.
• We want to select 4 novels and 2 biographies.
• The selected books are then arranged in a row on a shelf.
• Order of books on the shelf matters.

Concept / Approach:
The process has two stages. First we select the required books using combinations, because the selection itself does not care about order. After selection, we arrange the chosen books in all possible orders using permutations, because order on the shelf matters.

Step-by-Step Solution:
Step 1: Choose 4 novels out of 5 distinct novels. This can be done in 5C4 ways.Step 2: Compute 5C4 = 5.Step 3: Choose 2 biographies out of 4 distinct biographies. This can be done in 4C2 ways.Step 4: Compute 4C2 = 6.Step 5: Total ways to select the required 6 books (4 novels and 2 biographies) = 5C4 * 4C2 = 5 * 6 = 30.Step 6: After selection, we have 6 distinct books to place on the shelf.Step 7: Number of permutations of 6 distinct books = 6! = 720.Step 8: Total arrangements = number of selections * arrangements of each selection = 30 * 720 = 21600.

Verification / Alternative check:
You can sanity check by noting that if all 9 books were distinct and all 6 positions were free, the total permutations would be 9P6, which is larger than any option. Our value 21600 is noticeably smaller than 9P6 and consistent with first selecting, then arranging.

Why Other Options Are Wrong:

• 26100, 24000, and 36000 are nearby large numbers but do not match the exact computation of 30 * 720.
• They may result from mistakes such as using 9C6 instead of the correct selection structure or miscalculating factorial values.

Common Pitfalls:
Candidates often confuse combinations and permutations. Some simply compute 9C6 and then multiply by 6!, forgetting the selection constraint of exactly 4 novels and exactly 2 biographies. Others might add counts instead of multiplying them. Always separate the problem into selection and arrangement phases and handle each carefully.

Final Answer:
The number of ways to select and arrange 4 novels and 2 biographies is 21600, so the correct answer is 21600.