From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is:

Options

A. A. At least 500 but less than 750
B. B. At least 750 but less than 1000
C. C. At least 1000
D. D. Less than 500

Answer

To solve the problem step by step, we need to find the number of ways to select and arrange 4 novels and 1 dictionary such that the dictionary is always in the middle.

### Step 1: Selection of Novels and Dictionary
We have 6 different novels and 3 different dictionaries. We need to select:
- 4 novels from 6
- 1 dictionary from 3

The number of ways to select the novels can be calculated using the combination formula \( nCr \), which is given by:

\[ nCr = \frac{n!}{r!(n-r)!} \]

So, for selecting 4 novels from 6:

\[ \text{Number of ways to select novels} = 6C4 = \frac{6!}{4! \cdot (6-4)!} = \frac{6!}{4! \cdot 2!} \]

Calculating this:

\[ 6C4 = \frac{6 \times 5}{2 \times 1} = 15 \]

Next, for selecting 1 dictionary from 3:

\[ \text{Number of ways to select a dictionary} = 3C1 = \frac{3!}{1! \cdot (3-1)!} = \frac{3!}{1! \cdot 2!} = 3 \]

### Step 2: Arranging the Selected Items
After selecting 4 novels and 1 dictionary, we need to arrange them in a row such that the dictionary is always in the middle.

The arrangement will look like this:

\[ \text{Novel} \quad \text{Novel} \quad \text{Dictionary} \quad \text{Novel} \quad \text{Novel} \]

This means we have 4 positions for novels and 1 fixed position for the dictionary.

The number of ways to arrange the 4 novels in the 4 positions is given by:

\[ \text{Number of arrangements of novels} = 4! = 24 \]

### Step 3: Total Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to select the novels, the number of ways to select the dictionary, and the number of arrangements of the novels.

\[ \text{Total arrangements} = (\text{Ways to select novels}) \times (\text{Ways to select dictionary}) \times (\text{Arrangements of novels}) \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 15 \times 3 \times 24 \]

Calculating this:

\[ 15 \times 3 = 45 \]
\[ 45 \times 24 = 1080 \]

### Final Answer
Thus, the total number of arrangements is **1080**.