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?

To solve this, we need to find the number of non negative integer solutions for the equation 1S plus 4F plus 6X equals 25, where S is the number of singles, F is the number of fours, and X is the number of sixes.

The number of singles S is automatically determined once we choose the number of fours and sixes, as long as 4F plus 6X is less than or equal to 25. Therefore, we only need to find all possible combinations of F and X such that 4F plus 6X is less than or equal to 25.

We test possible values for X:

If X is 0: 4F can be 0, 4, 8, 12, 16, 20, or 24. This gives 7 possibilities for F (0 to 6).

If X is 1: 4F plus 6 is less than or equal to 25, so 4F is less than or equal to 19. 4F can be 0, 4, 8, 12, or 16. This gives 5 possibilities for F (0 to 4).

If X is 2: 4F plus 12 is less than or equal to 25, so 4F is less than or equal to 13. 4F can be 0, 4, 8, or 12. This gives 4 possibilities for F (0 to 3).

If X is 3: 4F plus 18 is less than or equal to 25, so 4F is less than or equal to 7. 4F can be 0 or 4. This gives 2 possibilities for F (0 or 1).

If X is 4: 4F plus 24 is less than or equal to 25, so 4F is less than or equal to 1. 4F can only be 0. This gives 1 possibility for F (0).

Total ways = 7 plus 5 plus 4 plus 2 plus 1 = 19.

The correct option is B.