A flag has to be designed with 4 horizontal stripes using some or all of the colours red, green and yellow. What is the number of different ways in which this can be done so that no two adjacent stripes have the same colour ?

(a) 12
(b) 18
(c) 24 ✓ UPSC's answer
(d) 36

Why the answer is (c)

• The flag consists of 4 horizontal stripes, and there are 3 available colors: red, green, and yellow.

• For the first stripe, any of the 3 colors can be chosen, giving 3 options.

• For the second stripe, the color must be different from the first stripe, leaving 2 options.

• For the third stripe, the color must be different from the second stripe, leaving 2 options.

• For the fourth stripe, the color must be different from the third stripe, leaving 2 options.

• The total number of ways is the product of these choices: 3 × 2 × 2 × 2 = 24.

Why the other options are wrong

(a) 12: 12 is incorrect because it likely assumes only 2 colors are used or miscalculates the permutations for the subsequent stripes.
(b) 18: 18 is incorrect because it does not match the multiplicative principle of 3 choices for the first stripe and 2 for each of the remaining three.
(d) 36: 36 is incorrect because it assumes 3 choices for every stripe (3^3 or similar error), ignoring the constraint that adjacent stripes cannot have the same color.

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.