How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
Solutions · 1
Official solution
This problem is solved using gap method. As here no 'S' is adjacent to each other so we have to put them in the gap. So first write all the letters other than 'S' such a way that there is a gap between two letters.
Given word is MISSISSIPPI.
Here, I = 4 times, S = 4 times, P = 2 times, M = 1 time
\_M\_I\_I\_I\_I\_P\_P\_
Those seven letters M, I, I, I, I, P, P can be arranged in \[{{7!} \over {4!2!}}\] ways
Those seven letters creates 8 gaps and we have to choose 4 gaps from those 8 gaps to put those four 'S' letters.
This can be done 8C4 ways.
After placing those four 'S' letters we can arrange them in \[{{4!} \over {4!}}\] ways.
Therefore, required number of words
\[ = {{7!} \over {4!2!}} \times {}^8{C_4} \times {{4!} \over {4!}}\]
\[ = {{7\,.\,6!} \over {4!4!}} \times {}^8{C_4}\]
\[ = 7\,.\,{}^6{C_4}\,.\,{}^8{C_4}\]