Correct Option (1)

To determine the number of distinct combinations for scoring exactly 25 runs using only singles (1 run), fours (4 runs), and sixes (6 runs), we formulate a linear Diophantine equation.

Let:

x = number of singles (1 run)
y = number of fours (4 runs)
z = number of sixes (6 runs)

The objective is to find the number of non-negative integer solutions (x, y, z) for the equation:

x + 4y + 6z = 25

We systematically enumerate all possible combinations by iterating through the maximum possible values for z (number of sixes), then y (number of fours), and subsequently calculating x (number of singles), ensuring x ≥ 0:

When z = 0 (no sixes): x + 4y = 25
- (25, 0, 0)
- (21, 1, 0)
- (17, 2, 0)
- (13, 3, 0)
- (9, 4, 0)
- (5, 5, 0)
- (1, 6, 0)
When z = 1 (one six): x + 4y = 19
- (19, 0, 1)
- (15, 1, 1)
- (11, 2, 1)
- (7, 3, 1)
- (3, 4, 1)
When z = 2 (two sixes): x + 4y = 13
- (13, 0, 2)
- (9, 1, 2)
- (5, 2, 2)
- (1, 3, 2)
When z = 3 (three sixes): x + 4y = 7
- (7, 0, 3)
- (3, 1, 3)

Summing these combinations yields a total of 7 + 5 + 4 + 2 = 18 distinct ways to score exactly 25 runs.

Incorrect Options:

Options 2 (19), 3 (20), and 4 (21) are incorrect. The systematic enumeration of all valid non-negative integer combinations for x, y, and z that satisfy the equation x + 4y + 6z = 25 results in a total of 18 distinct ways, as detailed in the correct option explanation.