How many permutations of the letters in the word ENGINEERING such that all E's are together and no N's are together?

Level: Senior

Ignoring the N's, each desired permutation has 6 letters, namely 1 block of E's, 2 G's, 2 I's and 1 R. There are 6!/(1!*2!*2!*1!) = 180 permutations of the letters in ENGINEERING with all E's together and without N's. For each of these permutations, the 6 letters, considering the block of E's as 1 superletter, created 7 gaps between them. Pictorially, for example, _EEE_G_G_I_I_R_ has 7 gaps (or underscores). There are 7C3 = 35 ways to fill the 3 N's into these 7 gaps. Hence the number of prescribed permutations is 180*35 = 6300.