Question

The number of permutations of the letters of the word CONSEQUENCE in which all the three E’s are together is
A) $9!3!$
B) $\dfrac{{9!}}{{2!2!}}$
C) $\dfrac{{9!}}{{2!2!3!}}$
D) $\dfrac{{9!}}{{2!3!}}$

Answer 1

In order to find the solution for a given problem you have to know the formula for permutation and how to implement it. Then count how many letters are present in CONSEQUENCE. After that you have to see how many E’s are there and how many combinations are possible to make with remaining letters. Now, just apply the formula of permutation and you will find the right answer.

Complete step by step answer:
First of all we have to see what is given word,
$\Rightarrow$ CONSEQUENCE
Now, let’s count how many total letters are there in CONSEQUENCE,
$\Rightarrow$ Total letters in CONSEQUENCE = $11$
Now, count individual letter,
$\Rightarrow$Total C = $2$
$\Rightarrow$Total E = $3$
$\Rightarrow$Total N = $2$
$\Rightarrow$Total O = $1$
$\Rightarrow$Total Q = $1$
$\Rightarrow$Total S = $1$
$\Rightarrow$Total U = $1$
Now, we have to find how many permutations are possible with three E’s together.
So, there are a total of three E’s and the remaining letters will be 8.
So, Total possible letters will be nine.
Now, apply formula of permutation,
Required permutation is,
$\Rightarrow \dfrac{{9!}}{{2!2!}}$
This denominator comes because there are 2 of C’s and 2 of N’s.
So, correct option is option (B) $\dfrac{{9!}}{{2!2!}}$.

Note:
Permutation and combination are the methods of counting which help us to determine the number of different ways of arranging and selecting objects out of a given number of objects, without listing them. In this problem we use permutation because they have asked for how many ways we can arrange words with three E's together.