# Q. 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

(c) 20

(d) 21

Correct Answer: (b) 19

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## In how many ways a batsman can score 25 runs ?

Let x = singles, y = fours, z = sixes

We need to find integer solutions to: x + 4y + 6z = 25

Approach: Systematically consider possible values for z (sixes) from maximum to minimum.

Solutions:

z = 4: (1,0,4) → 1 way (four sixes and one single)

z = 3: (7,0,3), (3,1,3) → 2 ways

z = 2: (13,0,2), (9,1,2), (5,2,2), (1,3,2) → 4 ways

z = 1: (19,0,1), (15,1,1), (11,2,1), (7,3,1), (3,4,1) → 5 ways

z = 0: (25,0,0), (21,1,0), (17,2,0), (13,3,0), (9,4,0), (5,5,0), (1,6,0) → 7 ways

Each tuple represents (singles, fours, sixes).

Total number of ways: 1 + 2 + 4 + 5 + 7 = 19

Therefore, there are 19 different ways for a batsman to score exactly 25 runs using combinations of singles, fours, and sixes.