A, B, C, D, E, F, G =7 letters gives 7! =5040 permutations. Out of the 5040 permutations, there are 52 permutations with the word "BEG" in them. If you exclude those 52 permutations, then you have:
Sorry, for the mistake in the number of "BEG" ! It is actually 120 permutations with the word "BEG" in them. I copied the wrong number. So, the total should be: 7! =5040 - 120 =4,920 permutations.
