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GRE Quant Problem Solving
1

I think I did this calculation correctly but wanted suggestions for a shorter way to do it. Can someone recommend a short way to do this?

My process:

Here I decided to first calculate the total number of ways GREGMAT can be arranged without any constraints. I arrived at 7! / 2! = 2520.

Then for the constraints, I calculated the following cases:

  1. What if the vowels are at both the ends. So, A_ _ _ _ _ E or E_ _ _ _ _ A.
  2. What if there’s a vowel only at the starting. So, A _ _ _ _ _ _ or E _ _ _ _ _ _.
  3. What if there’s a vowel only at the end. So, _ _ _ _ _ _ A or _ _ _ _ _ _ E.

For 1. the number of arrangements possible will be 2 x (5! / 2!) = 120
For 2. the number of arrangements possible will be 2 x (6! / 2!) = 720
For 3. the number of arrangements possible will be 2 x (6! / 2!) = 720

Since the cases for 1. are repeating in 2. and 3. as well, we can subtract 120 from 720 for each cases.

Final answer: 2520 - (120 + 720-120 + 720-120) = 1200.

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2

Assume momentarily that the two G's are distinct; then, there would be 2400 different possible ways under the established constraint. Now, you can divide by 2 since each arrangement is counted twice due to distinguishing between G_1 and G_2. This should lead you to your final answer of 1200.

Nevertheless, I think your way works just fine too.