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 arrangements is

Options

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

Answer

To solve the problem of selecting and arranging 4 novels and 1 dictionary such that the dictionary is always in the middle, we can follow these steps:

### Step 1: Understand the arrangement
The arrangement must be in the form of:
\[ \text{Novel, Novel, Dictionary, Novel, Novel} \]
This means that the dictionary will always occupy the middle position.

### Step 2: Select the novels and dictionary
We need to select:
- 4 novels from 6 different novels
- 1 dictionary from 3 different dictionaries

### Step 3: Calculate the combinations
The number of ways to choose 4 novels from 6 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
So, the number of ways to choose 4 novels from 6 is:
\[ \binom{6}{4} = \frac{6!}{4! \cdot (6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

The number of ways to choose 1 dictionary from 3 is:
\[ \binom{3}{1} = \frac{3!}{1! \cdot (3-1)!} = \frac{3!}{1! \cdot 2!} = 3 \]

### Step 4: Calculate arrangements of the selected novels
Once we have selected 4 novels, we can arrange these 4 novels in the 4 positions available (since the dictionary is fixed in the middle). The number of arrangements of 4 novels is given by:
\[ 4! = 24 \]

### Step 5: Combine all parts to find the total arrangements
Now, we multiply the number of ways to choose the novels, the number of ways to choose the dictionary, and the number of arrangements of the novels:
\[ \text{Total arrangements} = \binom{6}{4} \times \binom{3}{1} \times 4! = 15 \times 3 \times 24 \]

### Step 6: Calculate the final result
Calculating the above expression:
\[ 15 \times 3 = 45 \]
\[ 45 \times 24 = 1080 \]

Thus, the total number of arrangements is **1080**.

### Conclusion
The number of arrangements of 4 novels and 1 dictionary, with the dictionary always in the middle, is **1080**.

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