Explanation:

- We know \( p, q, r, s \) are in arithmetic progression (AP).

- The property of AP means: \( q = p + d \), \( r = p + 2d \), and \( s = p + 3d \), where \( d \) is the common difference.

- Given: \( p + s = 8 \).

- So, \( p + (p + 3d) = 8 \) => \( 2p + 3d = 8 \).

- Given: \( qr = 15 \).

- So, \( (p + d)(p + 2d) = 15 \).

To find the difference between the largest and smallest numbers:

- The largest number (\(s\)) is \( p + 3d \) and the smallest (\(p\)) is \(p\).

- Difference = \( (p + 3d) - p = 3d \).

Working out with equations:

1. From \( 2p + 3d = 8 \): express \( p = \frac{8 - 3d}{2} \).

2. Substitute in \( (p + d)(p + 2d) = 15 \) and solve for \( p \) and \( d \).

3. Find possible values of \( d \) and calculate the difference \(3d\).

After calculations, \( d = 2 \).

- Thus, the difference = \(3d = 6\).

Correct option is:

- Option 1: 6