Question

If the letters of the word "QUEUE" are arranged in a possible 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:In this question, we first need to write the letters in the given word in alphabetical order. Now, before the required word there will be letters starting with E , QE in the dictionary. So, first we need to find the numbers of words starting with E, QE, QUE using the permutations formula of arranging n things out of p like things given by \[\dfrac{n!}{p!}\] and add all of them.

Complete step-by-step answer:
Now, form the given word QUEUE we have the letters as follows in the alphabetical order
EEQUU
PERMUTATION: Each of the different arrangements which can be made by taking some or all of a number of things is called a permutation.
As we already know that the number of permutations of n things taken all at a time, in which p are alike of one kind and rest are different is given by
\[\Rightarrow \dfrac{n!}{p!}\]
Now, for the word QUEUE to come in the dictionary there will be some other words before it
 EEQUU
Now, let us find the number of words starting with E

E

Now, the remaining letters EQUU can be arranged in
\[\Rightarrow \dfrac{4!}{2!}\]
Now, on further simplification we get,
\[\Rightarrow 4\times 3\]
Thus, there will be 12 words starting with E
Now, the alphabet Q gets fixed in the first position and then we need to find the words starting with QE as they come before the word QUEUE

Q

E

Now, we need to arrange the letters EUU using the above formula
\[\Rightarrow \dfrac{3!}{2!}\]
Now, on further simplification we get,
\[\Rightarrow 3\]
Thus, there are 3 words starting with QE
Now, the letters QUE gets fixed in the first three positions and the remaining letters are EU

Q

U

E

Now, the number of arrangements possible are
\[\Rightarrow 2!\]
Here, the word QUEEU comes before QUEUE in the alphabetical order
Thus, there are 2 words possible.
Now, the rank of the word QUEUE is given by
\[\begin{align}
  & \Rightarrow 12+3+2 \\
 & \Rightarrow 17 \\
\end{align}\]
Hence, the correct option is (c).

Note:It is important to note that we do not find all the words starting with Q because the required word comes somewhere in between. So, we fix that position and then go for next word in alphabetical order and then again check the number of possible words that occur before the required one.It is also to be noted that to get the rank of a particular word we need to count all the words that are possible to come before it and the required word itself. Because neglecting the required word does not give its rank.