Find the number of ways a batsman can score a double century only in terms of…

Let x be the number of 4s and y be the number of 6s.

1. Each 4 gives 4 runs and each 6 gives 6 runs, so we need 4x + 6y = 200.

2. Divide by 2 to simplify: 2x + 3y = 100.

3. Solve for x: x = (100 − 3y)/2. For x to be an integer, 100 − 3y must be even, so y must be even.

4. Write y = 2k where k ≥ 0. Then x = (100 − 6k)/2 = 50 − 3k. We need x ≥ 0, so 50 − 3k ≥ 0 ⇒ k ≤ 16.

Thus k can be 0, 1, 2, …, 16, giving 17 valid pairs (x, y). So there are 17 ways to score exactly 200 runs using only 4s and 6s.