Graduate Management Admission Test (GMAT) Practice Test

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How many ways can 6 children be arranged if A and E cannot sit next to each other?

The correct answer is: 120

To find the number of ways to arrange 6 children, with the condition that A and E cannot sit next to each other, we can take a systematic approach. First, calculate the total number of arrangements of the 6 children without any restrictions. The number of arrangements is given by the factorial of the number of children, which is 6! (6 factorial). Calculating that: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Next, we need to calculate the number of arrangements where A and E are sitting together. To do this, we can treat A and E as a single unit or block. This effectively reduces the number of units to arrange from 6 to 5 (the block AE plus the other 4 children). Now, calculate the arrangements of these 5 units: 5! = 5 × 4 × 3 × 2 × 1 = 120. However, within the block AE, A and E can switch places (i.e., AE or EA), which gives us an additional factor of 2 for each arrangement of the block. Thus, the total number of arrangements where A and E are next to each other would