Question

how many ways can the letters in the word PHENOMENON be arranged, if the first letter must be a M and the last letter cannot be an O?

Answer 1

How many duplicated letters? 2 E's; 3 N's; 2 O's: EEHMNNNOOP, 10 letters.

The number of ways of arranging 10 letters is 10*9*8*7*...*1=10!=3628800.

But we have to account for the duplicates by dividing by the product of duplicate arrangements. 2 ways of arranging 2 E's and 2 O's and 6 for 3 N's=2*2*6=24. So we get 3628800/24=151200.

Now we apply the constraints.

First letter must be M, so there are 9 letters left: 9!=362880. Divide by 24 to allow for duplicate E's, O's and N's: 362880/12=15120. But we mustn't terminate with O. 2/9 of these end in O, because of the 9 letters, 2 are O's, so the total is 11760, being 7/9*15120.