In how many different ways can 8 examination papers be arranged in a row , so that the best and the worst papers may never come together ?

Options

A. A. 30240
B. B. 23400
C. C. 12340
D. D. none of these

Answer

To solve the problem of arranging 8 examination papers in a row such that the best and worst papers never come together, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate Total Arrangements**:
The total number of ways to arrange 8 examination papers is given by the factorial of the number of papers.
\[ \text{Total arrangements} = 8! = 40320 \]

2. **Consider Best and Worst Papers Together**:
To find the arrangements where the best and worst papers are together, we can treat them as a single unit or block. This means we now have 7 units to arrange (the block of best and worst papers + 6 other papers).
\[ \text{Arrangements with best and worst together} = 7! = 5040 \]

3. **Arrangements of Best and Worst within the Block**:
The best and worst papers can be arranged within their block in 2 ways (best first or worst first).
\[ \text{Arrangements of best and worst papers} = 2! = 2 \]

4. **Total Arrangements with Best and Worst Together**:
Now, we multiply the arrangements of the 7 units by the arrangements of the best and worst papers within their block.
\[ \text{Total arrangements with best and worst together} = 7! \times 2! = 5040 \times 2 = 10080 \]

5. **Calculate Arrangements Where Best and Worst Never Come Together**:
To find the arrangements where the best and worst papers never come together, we subtract the arrangements where they are together from the total arrangements.
\[ \text{Arrangements where best and worst never together} = 8! - (7! \times 2!) = 40320 - 10080 = 30240 \]

### Final Answer:
The number of different ways to arrange the 8 examination papers such that the best and worst papers never come together is **30240**.

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