Question

How many different words can be formed from the letters of ‘CONSTANTINOPLE’ in which all three N come together?
(A) $\dfrac{{14!}}{{2!3!2!}}$
(B) $\dfrac{{12!}}{{2!2!}}$
(C) $\dfrac{{12!}}{{2!3!2!}}$
(D) None of these.

Answer 1

Hint: In this question, we have to find the number of arrangements possible of a given word with the conditions given. In the word given, there are some repeated letters, so we consider them as the same type of letters and we know that for a word having a total $n$ number of letters with $p$ number of letters exactly same kind and $q$ number of the letters are exactly same of another kind, then the arrangements are $\dfrac{{n!}}{{p!q!}}$

Complete step-by-step answer:
Given:
The word given is,
‘CONSTANTINOPLE’
The total number of letters in this word are 14.
In this word, the letter N is repeated 3 times, the letter O is repeated 2 times and the letter T is repeated 2 times.
Now, according to the question, we combine all three N together and take it as one letter.
After taking all three N together, the number of letters left in the word ‘CONSTANTINOPLE’ is 11.
So now, the total letters in the word ‘CONSTANTINOPLE’ are
$
 = {\text{ All three N combined as one + Rest of the letters in the word}}\\
{\text{ = }}1 + 11\\
 = 12
$
Then, the number of arrangements of 12 letters$ = 12!$
The number of times letter O is repeated is 2.
So, the number of arrangements of letter O ${\text{ = 2!}}$
Similarly, the number of times letter T is repeated is2.
So, the number of arrangements of letter T ${\text{ = 2!}}$
Now using the formula, the number of the words that can be formed from the letters of ‘CONSTANTINOPLE’ in which all three N come together is –
$ = \dfrac{{{\text{Number of arrangements of 12 letters}}}}{{{\text{number of arrangements of letter O }} \times {\text{ number of arrangements of letter T }}}}$
Substituting the values in above equation, we get,
$ = \dfrac{{12!}}{{2!2!}}$
Therefore, the correct option is (B) $\dfrac{{12!}}{{2!2!}}$

So, the correct answer is “Option B”.

Note: It should be noted that when we combine all three N together, it becomes one letter, that is why it was not considered as the repeated letter such as letter O and letter T. If we do not combine all three N together then the solution of this question can be given as-
The number of the words that can be formed from the letters of ‘CONSTANTINOPLE’$ = \dfrac{{14!}}{{2!3!2!}}$