Question

Total number of ways in which the letters of the word ‘MISSISSIPPI’ can be arranged, so that any two S’s are separated is equal to
A.7350
B.3650
C.6250
D.1261

Answer 1

Hint: In the word MISSISSIPPI, there are four I’s, four S’s, two P’s and one M. We have to find the number of ways in which no two S’s are separated, that is they are not placed together. Using the concept of permutation and combination we can identify the formula that has to be used here and thus get the correct answer.

Complete step-by-step answer:
The word can be written as _M_I_I_I_I_P_P_ and S can take eight positions represented as spaces between the letters. The rest of the seven letters can be arranged in $ \dfrac{{7!}}{{4!2!}} $ ways.
The term $ 7! $ is divided by $ 4!2! $ as the ‘I’ is repeated four times and the ‘P’ is repeated two times.
The number of ways in which the ‘S’ can be arranged so that none of them come together is $ \dfrac{{n!}}{{r!\left( {n - r} \right)!}} = \dfrac{{8!}}{{4!\left( {8 - 4} \right)!}} = \dfrac{{8!}}{{4!4!}} $
So, the total number of ways in which the letters of the word ‘MISSISSIPPI’ can be arranged so that any two S’s are separated is $ \dfrac{{8!}}{{4!4!}} \times \dfrac{{7!}}{{4!2!}} = \dfrac{{8 \times 7 \times 6 \times 5 \times 4!}}{{4!4!}} \times \dfrac{{7 \times 6 \times 5 \times 4!}}{{4!2!}} = \dfrac{{352800}}{{48}} = 7350 $
So, the correct answer is “Option A”.

Note: Arrangement is simply the grouping of objects according to the requirement. Permutation and Combination help us to find out the number of ways in which the objects can be arranged. In permutation, we consider the arrangement of objects in a specific order while in combination we arrange things without considering their order. In this question, we had to find only the number of ways in which the letters are arranged irrespective of their order, that’s why we use the formula of combination here.