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?

Question

Aicharya · Accepted Answer

We need non-negative integers $s$, $f$, $x$ with $s + 4f + 6x = 25$. For $x = 0$: $4f \le 25$, $f = 0$ to $6$, giving 7 ways. For $x = 1$: $4f \le 19$, $f = 0$ to $4$, giving 5 ways. For $x = 2$: $4f \le 13$, $f = 0$ to $3$, giving 4 ways. For $x = 3$: $4f \le 7$, $f = 0$ to $1$, giving 2 ways. For $x = 4$: $4f \le 1$, $f = 0$, giving 1 way. Total $= 7 + 5 + 4 + 2 + 1 = 19$.

In how many ways can a batsman score exactly 25 runs by scoring single runs, fours and sixes only,…

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