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?

The problem asks for the number of arrangements of 2 boys and 2 girls in a row such that the girls are not next to each other. We can solve this using the complementary counting principle: Total arrangements - Arrangements where girls ARE next to each other = Arrangements where girls are NOT next to each other. 1. Calculate the total number of arrangements: There are 4 distinct people (2 boys and 2 girls). The total number of ways to arrange them in a row is 4! (4 factorial). 4! = 4 x 3 x 2 x 1 = 24. 2. Calculate the number of arrangements where the two girls are next to each other: Treat the two girls as a single unit (GG). Now we are arranging 3 units: Boy1, Boy2, and (GG). These 3 units can be arranged in 3! ways = 3 x 2 x 1 = 6 ways. Within the (GG) unit, the two girls can swap positions (Girl1 Girl2 or Girl2 Girl1). So, there are 2! = 2 ways to arrange the girls within their block. Therefore, the number of arrangements where the girls are together is 3! * 2! = 6 * 2 = 12. 3. Subtract to find the desired arrangements: Number of arrangements where girls are NOT next to each other = Total arrangements - Arrangements where girls ARE next to each other = 24 - 12 = 12. Alternatively, by placing boys first: 1. Arrange the 2 boys: B B. There are 2! = 2 ways to arrange them (e.g., B1B2 or B2B1). 2. Now, create spaces around and between the boys: _ B _ B _. There are 3 possible spaces (indicated by underscores) where the girls can be placed so they are not next to each other. 3. We need to choose 2 of these 3 spaces for the 2 girls. This can be done in C(3, 2) = 3 ways. 4. Once the 2 spaces are chosen, the 2 girls can be arranged in those 2 spaces in 2! = 2 ways. 5. So, for each arrangement of boys, there are C(3, 2) * 2! = 3 * 2 = 6 ways to place the girls. 6. Total arrangements = (Ways to arrange boys) * (Ways to place girls) = 2 * 6 = 12. Both methods yield 12. Options analysis: A) 3: Incorrect. B) 6: Incorrect. This would be the answer if the girls were identical and placed around one specific arrangement of boys, or if there were fewer constraints. C) 12: Correct. D) 24: Incorrect. This is the total number of arrangements without any restrictions. The final answer is C.