Question

If all the letters of the word “QUEUE” are arranged in all manner as they are in a dictionary, then the rank of the word QUEUE is:
a) \[{{15}^{th}}\]
b) \[{{16}^{th}}\]
c) \[{{17}^{th}}\]
d) \[{{18}^{th}}\]

Answer 1

Hint: We need to find the rank of word QUEUE in the list of words formed by letters of the word QUEUE. So, firstly we need to find the words that would come before the word QUEUE in the list. Those words would consist of letter E at starting or the letter Q followed by letter E at the second position. Then we need to find how many words are there which have Q as the first letter and U as the second letter. Hence, add all the numbers u get to find the rank of the word QUEUE in the list.

Complete step by step answer:
We have the given word: QUEUE
According to the dictionary, the word starting with E should be at first place. So, the words with E as starting letter are: \[\text{E }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\] where, we have to choose from remaining four letters. So, the number of words with starting letter E are: $\dfrac{4!}{2!}=12$
Later, we have Q as a starting letter. So, again according to the dictionary, for second place E should come first. Therefore, the words with Q as first letter and E as second letter are: $\text{Q E }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }$ where, we have to choose from remaining three letters.
So, the number of the words with Q as the first letter and E as the second letter is $\dfrac{3!}{2!}=3$
After that, we need to take U as the second letter with Q as the starting letter. So, the words with U as second letter are: $\text{Q U }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }$ where, we have to choose from remaining three letters.
So, the number of the words with Q as the first letter and U as the second letter is $\dfrac{3!}{2!}=3$
But we need to find the rank of QUEUE.
So, we need to find the number of words with Q as the starting letter, U as the second letter, and E as the third letter, i.e. $\text{Q U E }\!\!\_\!\!\text{ }\!\!\_\!\!\text{ }$ where we have to choose from the remaining two letters.
So, the number of the words with Q as the first letter and U as the second letter and E as the third letter is: $2!=2$
The words with Q as the starting letter, U as the second letter, and E as the third letter would be QUEEU and QUEUE.
Hence the rank of QUEUE in the list is $12+3+2=17$

So, the correct answer is “Option C”.

Note: Since the letter E and U are repeated twice in the word, so be careful to divide the answer by 2!
 Also, as we state that the word consists of Q as a starting letter, so the first place of the word is occupied. We only need to fill the remaining 4 letters. So, be careful when filling up the places. Some might write it as 5! Instead of 4!