In how many can the lettes in English alphabet be arranged, so that there are 7 letters between the letters a and b.

Answer

To solve the problem of arranging the letters in the English alphabet such that there are 7 letters between the letters A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Positions of A and B**:
- We need to place A and B such that there are exactly 7 letters between them. This means if A is in position \( x \), then B must be in position \( x + 8 \) (since A occupies one position and there are 7 letters in between).
- The positions of A and B can be represented as follows:
- If A is in position 1, B will be in position 9.
- If A is in position 2, B will be in position 10.
- Continuing this way, if A is in position 18, B will be in position 26.

2. **Calculate the Possible Positions**:
- The valid positions for A range from 1 to 18 (because if A is in position 18, B will be in position 26, which is the last position).
- Therefore, there are 18 possible positions for A (and correspondingly for B).

3. **Arrange the Remaining Letters**:
- After placing A and B, we have 24 letters left (since there are 26 letters in total and A and B are fixed).
- These 24 letters can be arranged in any order in the remaining 24 positions.

4. **Calculate the Arrangements of the Remaining Letters**:
- The number of ways to arrange the 24 remaining letters is given by \( 24! \) (24 factorial).

5. **Consider the Order of A and B**:
- Since A and B can be arranged in two ways (A can be first or B can be first), we need to multiply our result by 2.

6. **Combine All Parts**:
- The total number of arrangements is given by:
\[ \text{Total Arrangements} = (\text{Number of positions for A and B}) \times (\text{Arrangements of remaining letters}) \times (\text{Order of A and B}) \]
- This can be expressed mathematically as:
\[ \text{Total Arrangements} = 18 \times 24! \times 2 \]

### Final Answer:
Thus, the total number of ways to arrange the letters such that there are 7 letters between A and B is:
\[ \text{Total Arrangements} = 36 \times 24! \]