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?

1. A. 18
2. B. 19
3. C. 20
4. D. 21

Answer: B

Explanation

Let x be the number of singles, y be the number of fours, and z be the number of sixes. The problem requires finding the number of non-negative integer solutions to the equation x + 4y + 6z = 25. We can systematically find combinations by iterating through possible values of z (number of sixes):
1. If z = 0: x + 4y = 25. Possible (x, y) pairs are (1, 6), (5, 5), (9, 4), (13, 3), (17, 2), (21, 1), (25, 0). (7 ways)
2. If z = 1: x + 4y = 19. Possible (x, y) pairs are (3, 4), (7, 3), (11, 2), (15, 1), (19, 0). (5 ways)
3. If z = 2: x + 4y = 13. Possible (x, y) pairs are (1, 3), (5, 2), (9, 1), (13, 0). (4 ways)
4. If z = 3: x + 4y = 7. Possible (x, y) pairs are (3, 1), (7, 0). (2 ways)
5. If z = 4: x + 4y = 1. Possible (x, y) pair is (1, 0). (1 way)
For z ≥ 5, 6z would be 30 or more, exceeding 25, so no solutions exist. Summing the ways from each case: 7 + 5 + 4 + 2 + 1 = 19 ways. This question tests basic combinatorics and systematic enumeration, a common type in CSAT.