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 (1) less than 500 (2) at least 500 but less than 750 (3) at least 750 but less than 1000 (4) at least 1000

Options

A. atleast 1000
B. less than 500
C. atleast 500 but less than 750
D. atleast 750 but less than 1000

Explanation

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: Select the Novels
We need to select 4 novels from the 6 different novels available. The number of ways to choose 4 novels from 6 is given by the combination formula:

\[ \text{Number of ways to select 4 novels} = \binom{6}{4} \]

### Step 2: Calculate the Combinations
Using the combination formula:

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

We can calculate:

\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Step 3: Arrange the Selected Novels
Once we have selected 4 novels, we need to arrange them. The number of ways to arrange 4 novels is given by:

\[ 4! = 24 \]

### Step 4: Select the Dictionary
Next, we need to select 1 dictionary from the 3 different dictionaries available. The number of ways to choose 1 dictionary is:

\[ \text{Number of ways to select 1 dictionary} = \binom{3}{1} = 3 \]

### Step 5: Total Arrangements
Now, we can calculate the total number of arrangements. Since the dictionary is always in the middle, the arrangement will be of the form: Novel, Novel, Dictionary, Novel, Novel.

Thus, the total number of arrangements is:

\[ \text{Total arrangements} = \binom{6}{4} \times 4! \times \binom{3}{1} \]

Substituting the values we calculated:

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

### Step 6: Calculate the Final Answer
Now we can compute:

\[ 15 \times 24 = 360 \]
\[ 360 \times 3 = 1080 \]

### Conclusion
The total number of arrangements is 1080. Therefore, the answer is:

**(4) at least 1000.**

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