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 ?
- (a) 18
- (b) 19 ✓ UPSC's answer
- (c) 20
- (d) 21
Why the answer is (b)
• Let s, f and x be the numbers of singles, fours and sixes, so s + 4f + 6x = 25 with each at least 0; singles fill whatever is left, so every valid pair (f, x) gives exactly one way.
• x = 0: 4f ≤ 25, so f = 0 to 6 — 7 ways.
• x = 1: 4f ≤ 19, so f = 0 to 4 — 5 ways.
• x = 2: 4f ≤ 13, so f = 0 to 3 — 4 ways.
• x = 3: 4f ≤ 7, so f = 0 or 1 — 2 ways.
• x = 4: 4f ≤ 1, so f = 0 — 1 way.
• Total = 7 + 5 + 4 + 2 + 1 = 19, option (b).
Why the other options are wrong
- (a) 18
- 18 misses the case with no sixes, six fours and one single.
- (c) 20
- There are only 19 combinations of fours and sixes, not 20.
- (d) 21
- There are only 19 combinations of fours and sixes, not 21.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.