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?
Answer & explanation
Answer: (b) 19
Count the combinations of sixes, fours and singles that add up to 25; the order of shots does not matter. Fix the number of sixes, count how many fours can fit, and let singles make up the rest: 7 + 5 + 4 + 2 + 1 = 19.
- Let the batsman hit x sixes, y fours and z singles: 6x + 4y + z = 25. Once x and y are chosen, z is fixed.
- x = 0: 4y ≤ 25, so y = 0 to 6 → 7 ways.
- x = 1: 4y ≤ 19, so y = 0 to 4 → 5 ways.
- x = 2: 4y ≤ 13, so y = 0 to 3 → 4 ways.
- x = 3: 4y ≤ 7, so y = 0 or 1 → 2 ways.
- x = 4: 4y ≤ 1, so y = 0 → 1 way. Total = 7 + 5 + 4 + 2 + 1 = 19.
Remember · For 'irrespective of order' counts, fix the biggest unit, count options for the next; the smallest unit fills the rest automatically.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·