If the nine letters A, S, S, E, M, B, L, E, D are arranged in a row such that the two letters S, S, are together and the other vowels A, E, E, are separated by one or more letters, how many different such arrangements are possible?
There are eight possible places for the two S's. We must go through each of these cases one at a time.
Say the S's are placed in the first and second slots. Then there are 10 possible places for the vowels (slots 357, 358, 359, 368, 369, 379, 468, 469, 479, and 579) and three possible arrangements for each of these places, so there are a total of 30 possible arrangements of the vowels. Then there are six possible arrangements of M, B, and L, which must go into the remaining slots. Thus, there are a total of 180 possible arrangements corresponding to this case.
By symmetry, there are also 180 possible arrangements if the ...
We count the number of ways of arranging the letters in ASSEMBLED in such a way that the two S's are adjacent and no two vowels are adjacent.