Question

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
A. $6.\,{7.^8}{C_4}$
B. $6.\,{8.^7}{C_4}$
C. ${7.^8}{C_4}.{\,^6}{C_4}$
D. ${8.^8}{C_4}{\,^9}{C_4}$

Answer 1

Hint: Firstly, we will find the repeating letters. Further, we will find the permutation of S, single letter and then we will find the permutation of other than S letter. Therefore, we will multiply both of them to get the answer.

Complete step-by-step answer:
The word MISSISSIPPI. There are $11$ letters in the MISSISSIPPI words.
A part of that S is repeating $4$times, I is repeating $4$times, P is repeating $2$times and M is $1$time only.
Other than S, there are seven letters left M I I I P P I.
Now, $4$ S as a single letter, then the number of letters in the word MISSISSIPPI becomes $8$
So, total number of words can be arranged in the formed $ = {\,^8}{C_4}$
Now, other than S,
Seven letters MIIIPPI can arranged the letter in which I is repeating $4$ times and P is repeating 2 times
So, $\dfrac{{7!}}{{2!\,4!}}$
Therefore, number of words in which no two S are adjacent \[ = {\,^8}{C_4} \times \dfrac{{7!}}{{2!\,4!}}\]
$\Rightarrow$ Number of words in which no two S are adjacent \[ = {\,^8}{C_4} \times \dfrac{{7\,6!}}{{2!\,4!}}\]
$\Rightarrow$ Number of words in which no two S are adjacent \[ = {\,^8}{C_4} \times \dfrac{{7\,6!}}{{2!\,4!}}\]
$\Rightarrow$ Number of words in which no two S are adjacent $ = 7.{\,^8}{C_4}.{\,^6}{C_4}$
Thus, the number of words in which no two S are adjacent $ = 7.{\,^8}{C_4}.{\,^6}{C_4}$
So, the correct answer is “Option C”.

Note: Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor.