Question

There are 6 different novels and 3 different poetry books on a table. If 4 novels and 1 poetry book are to be selected and arranged in a row on a shelf such that the poetry book is always in the middle, then the number of such possible arrangements is:

• A. 270
• B. 180
• C. 540
• D. 1080

Hint:

When arranging objects with specific conditions (such as a book always being in the middle), treat the object as fixed and then arrange the other objects around it.

Answer 1

The Correct Option is D

We are given that there are 6 different novels and 3 different poetry books. We need to select 4 novels and 1 poetry book, and arrange them in a row on a shelf with the condition that the poetry book is always in the middle. Step 1: Arranging the poetry book in the middle Since the poetry book must always be in the middle, we have only 1 choice for the position of the poetry book. There is only 1 position for the poetry book in the middle of the 5 positions on the shelf. Step 2: Selecting and arranging the novels We need to select 4 novels from the 6 available novels. The number of ways to choose 4 novels from 6 is given by the combination formula: \[ \binom{6}{4} = \frac{6!}{4!(6 - 4)!} = \frac{6 \times 5}{2 \times 1} = 15. \] After selecting the 4 novels, we can arrange them in the 4 remaining positions. The number of ways to arrange 4 novels is \( 4! \), which is: \[ 4! = 4 \times 3 \times 2 \times 1 = 24. \] Step 3: Selecting the poetry book Since there are 3 different poetry books, we can choose any 1 of them in 3 ways. Step 4: Calculating the total number of arrangements The total number of arrangements is given by: \[ \binom{6}{4} \times 4! \times 3 = 15 \times 24 \times 3 = 1080. \] Thus, the total number of possible arrangements is \( 1080 \).

Answer 2

Problem: There are 6 different novels and 3 different poetry books on a table. We want to select 4 novels and 1 poetry book and arrange them in a row on a shelf such that the poetry book is always in the middle. Find the total number of such possible arrangements.

Step 1: Select the books - Select 4 novels out of 6: \[ \binom{6}{4} = 15. \] - Select 1 poetry book out of 3: \[ \binom{3}{1} = 3. \]

Step 2: Arrange the selected books - The poetry book must be placed in the middle (3rd position). - Arrange the 4 novels in the remaining 4 positions (1st, 2nd, 4th, and 5th). - Number of ways to arrange 4 novels in 4 positions: \[ 4! = 24. \]

Step 3: Calculate total arrangements \[ \text{Total} = \binom{6}{4} \times \binom{3}{1} \times 4! = 15 \times 3 \times 24 = 1080. \]

Final answer: \[ \boxed{1080}. \]