In how many ways can a batsman score exactly 25 runs by scoring single runs, fours and sixes only, irrespective of the sequence of scoring shots?

EXPLANATION

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.