Question

Consider the word W=MISSISSIPPI. Numbers of ways in which the letters of the word W can be arranged if at least one vowel is separated from the rest of the vowels
A.$\dfrac{{8!.161}}{{4!.4!.2!}}$
B.$\dfrac{{8!.161}}{{4.4!.2!}}$
C.$\dfrac{{8!.161}}{{4!.2!}}$
D.$\dfrac{{8!}}{{4!.2!}}.\dfrac{{165}}{{4!}}$

Answer 1

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 the letters of the word W=MISSISSIPPI can be arranged if at least one vowel is separated from rest of the vowels. 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 answer:
W=MISSISSIPPI, Letters = $11$
Four I’s, four S’s, two P’s and one M
Total number of ways arranging these letters = $\dfrac{{11!}}{{4!.4!.2!}}$
Suppose all I’s are together, then the number of ways = $\dfrac{{\left( {1 + 1 + 4 + 2} \right)!}}{{4!.2!}}$
(As here we have four I’s, we will consider these four I’s as one letter. Apart from this we have one M, four S’s and two P’s)
$= \dfrac{{8!}}{{4!.2!}}$
Therefore, the number of ways in which atleast one vowel is separated
$= \dfrac{{11!}}{{4!.4!.2!}} - \dfrac{{8!}}{{4!.2!}}$
Now we will expand $11!$ upto $8!$
$= \dfrac{{11 \times 10 \times 9 \times 8!}}{{4!.4!.2!}} - \dfrac{{8!}}{{4!.2!}}$
$= \dfrac{{11 \times 10 \times 9 \times 8!}}{{4!.4!.2!}} - \dfrac{{8!}}{{4!.2!}}.\dfrac{{4!}}{{4!}}$
Now we will take $\dfrac{{8!}}{{4!.4!.2!}}$ as a common factor.
$= \dfrac{{8!}}{{4!.4!.2!}}\left[ {11 \times 10 \times 9 - 4!} \right]$
On multiplication of terms written inside the bracket and expanding $4!$, we get
$= \dfrac{{8!}}{{4!.4!.2!}}\left[ {990 - 24} \right]$
On subtraction of terms written inside the bracket, we get
$= \dfrac{{8!}}{{4!.4!.2!}} \times 966$
On expanding $4!$ (written in denominator), we get
$= \dfrac{{8!}}{{4!.2!}} \times \dfrac{{966}}{{24}}$
On division, we get
$= \dfrac{{8!}}{{4!.2!}} \times \dfrac{{161}}{4}$
$= \dfrac{{8!.161}}{{4.4!.2!}}$

Therefore, the correct option is B.

Note:
The 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.