Graduate Management Admission Test (GMAT) Practice Test

How many different arrangements are there for 6 children where two specific children cannot sit next to each other?

1. 600

2. 720

3. 540

4. 840

The correct answer is: 720

To determine the number of different arrangements for 6 children such that two specific children do not sit next to each other, it's helpful to use the principle of complementary counting. First, calculate the total arrangements of 6 children without any restrictions. The total number of arrangements is given by the factorial of the number of children, which is 6! (6 factorial). This equals 720, as it is calculated as follows: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Next, consider the arrangements where the two specific children sit next to each other. If we treat these two children as a single unit or block, we effectively reduce the problem to arranging 5 units: the block of the two children and the other 4 individual children. The number of arrangements for these 5 units is 5!, which equals: 5! = 5 × 4 × 3 × 2 × 1 = 120. However, within their block, the two specific children can switch positions, adding another factor of 2 for their arrangement. Therefore, the total arrangements where these two children are sitting next to each other is: 5! × 2 = 120