Question

The number of arrangements that can be made out the letters of the word 'MISSISSIPI' so that all the S's come together and I's not come together is
A. $ 188 $
B. $ 186 $
C. $ 185 $
D. $ 190 $

Answer 1

Hint: Here in this question, first of all find the total number of arrangements in which all S’s are together then find the total number of arrangements in which all S’s and all I’s are together and then find subtract them so that you will get the required number of arrangements in which all S’s will be together and I’s will not be together.
and I’s will not be together.
Formula used: Total ways of arranging “n” things in which “P” and “Q” things are alike: $ \dfrac{{n!}}{{P!Q!}} $

Complete step-by-step answer:
We will find the total number of arrangements with given conditions, by finding first the total number of arrangements where S’s are together, that will be given as
Since the word “MISSISSIPI” have four “S” and four “I”, here we have $ 7 $ to be arranged because all S’s are together
So,
 $ n\left( {{A_1}} \right) = \dfrac{{7!}}{{4!4!}} \times 4! = \dfrac{{7!}}{{4!}} $
Now, we will find the total number of arrangements in which all I’s and all S’s are together,
In this case we have only $ 4 $ letters to be arranged, because we are considering all I’s and all S’s are together, so
 $ n\left( {{A_2}} \right) = \dfrac{{4!}}{{4!4!}} \times 4! \times 4! = 4! $
Therefore number of arrangements in which all S’s are together and I’s are not together will be given as
 $
  n\left( A \right) = n\left( {{A_1}} \right) - n\left( {{A_2}} \right) \\
   = \dfrac{{7!}}{{4!}} - 4! \\
   = \dfrac{{7 \times 6 \times 5 \times 4!}}{{4!}} - 4! \\
   = 210 - 24 \\
   = 186 \;
  $
So option B. is correct.
So, the correct answer is “Option B”.

Note: Sometimes finding the number of ways with a given set of conditions is a bit difficult by general formulas, so we have to find ourselves a better way of tackling or solving the condition like in this question and then use general formulas to evaluate further.