`P = {1,2,3,4,5……25}`. How many sets of integers can you pick from the set P so that they start with 1, end with 25 and consecutive numbers in the sets have a constant gap between them ?
For example , one way of selecting such a set of numbers in `{1,7,13,19,25}` (since the set starts with 1 , ends in 25 and consecutive numbers are at a constant gap of 6).

Options

A. 4
B. 5
C. 8
D. 10

Answer

To solve the problem, we need to determine how many sets of integers can be formed from the set \( P = \{1, 2, 3, \ldots, 25\} \) such that the sets start with 1, end with 25, and have a constant gap \( D \) between consecutive numbers.

### Step-by-Step Solution:

1. **Understand the Arithmetic Progression (AP)**:
- The first term \( A_1 = 1 \).
- The last term \( A_n = 25 \).
- The common difference between consecutive terms is \( D \).
- The \( n \)-th term of an AP can be expressed as:
\[ A_n = A_1 + (n - 1)D \]
- Substituting the known values:
\[ 25 = 1 + (n - 1)D \]

2. **Rearranging the Equation**:
- Rearranging gives:
\[ 25 - 1 = (n - 1)D \implies 24 = (n - 1)D \]

3. **Finding Factor Pairs**:
- We need to find pairs of integers \( (n - 1) \) and \( D \) such that their product equals 24.
- The factor pairs of 24 are:
- \( 1 \times 24 \)
- \( 2 \times 12 \)
- \( 3 \times 8 \)
- \( 4 \times 6 \)
- \( 6 \times 4 \)
- \( 8 \times 3 \)
- \( 12 \times 2 \)
- \( 24 \times 1 \)

4. **Counting Valid Sets**:
- Each factor pair \( (n - 1, D) \) corresponds to a valid set. The values of \( n - 1 \) can be derived from each factor:
- \( (1, 24) \) gives \( n = 2 \)
- \( (2, 12) \) gives \( n = 3 \)
- \( (3, 8) \) gives \( n = 4 \)
- \( (4, 6) \) gives \( n = 5 \)
- \( (6, 4) \) gives \( n = 7 \)
- \( (8, 3) \) gives \( n = 9 \)
- \( (12, 2) \) gives \( n = 13 \)
- \( (24, 1) \) gives \( n = 25 \)

5. **Total Combinations**:
- Each of the 8 factor pairs gives a valid configuration for the sets. Thus, there are a total of 8 sets that can be formed under the given conditions.

### Final Answer:
The total number of sets of integers that can be picked from the set \( P \) is **8**.