If 2 boys and 2 girls are to be arranged in a row so that the girls are not next to each other, how many possible arrangements are there?

(a) 3
(b) 6
(c) 12 ✓ UPSC's answer
(d) 24

Why the answer is (c)

• First, arrange the 2 boys in a row, which can be done in 2! = 2 ways.

• Placing the 2 boys creates 3 possible spaces (before the first boy, between the boys, and after the second boy) where the girls can be placed to ensure they are not adjacent.

• Select 2 out of these 3 spaces for the girls, which can be done in 3P2 = 3 × 2 = 6 ways.

• Alternatively, place the girls in the selected spaces in 2! = 2 ways, but since the selection of spaces already accounts for order in permutation logic, we simply multiply the boy arrangements by the number of valid girl placements: 2 (boys) × 3 (choices of spaces) × 2 (arrangement of girls in those spaces) is incorrect; the correct logic is 2! (boys) × 3P2 (girls in spaces) = 2 × 6 = 12.

• Therefore, the total number of possible arrangements is 12.

Why the other options are wrong

(a) 3: 3 is the number of ways to choose the spaces for the girls, not the total arrangements including the boys' permutations.
(b) 6: 6 is the number of ways to arrange the girls in the spaces or the total permutations of 3 items, ignoring the boys' distinct arrangements.
(d) 24: 24 is the total number of unrestricted arrangements of 4 distinct people (4!), which includes cases where the girls are adjacent.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2017, held on 18 June 2017. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.