Question

If the letters of the word “VARUN” are written in all possible ways and then are arranged as in dictionary, then the rank of the word “VARUN” is:
(a) 98
(b) 99
(c) 100
(d) 101

Answer 1

Hint: To this question, firstly we will calculate the total number of words starting with letter A,N,R and U as they appear before V in alphabetic order. After that, we will find the number of words with fixed letter arrangement of VAN as R appears after word N, and the we calculate the number of words with fixed arrangement of VARN as N appears before letter U. then, we will add total number of letters which occur before word “VARUN”, and then w will find rank by adding “1” to the total number of letters occurring before “VARUN”.

Complete step-by-step answer:
Now, in the word “VARUN”, we have a total five alphabets as V, A, R, U and N.
So, if we arrange the letters of word “VARUN”, alphabetically, then the first word which will be form is
A
N
R
U
V
Now, so in alphabetical order, the words starting with letter A will come first, then words with letter N then with letter R, then with letter U and then with letter V.
So, the total number of words starting with letter A will be equal to 4! as A is fixed at first position and we are left with 4 letters and these four letters can be rearranged in 4! Ways.
In same way, total number of words starting with letter N will be equals to 4!,
total number of words starting with letter R will be equals to 4!,
total number of words starting with letter U will be equals to 4!,
Now, for words starting with, the first word will be VANRU.
But for the word “VARUN”, R comes after N, so the number of words before the words starting with VAR, will be equal to 2! as VAN is fixed and we can arrange U and R in 2! Ways.
Now, in alphabetical order, N comes before U, so the number of words before words starting with VARU, will be equal to 1! as VARN is fixed and we can arrange U 1! Ways.
And next word after VARNU will be VARUN,
Then, the total number of words occurring before the word “VARUN” will be equal to 4!+ 4!+ 4!+ 4!+ 2!+ 1!
We know that, factorial function, $n!=n(n-1)(n-2).....3.2.1$, then
$=4\times (4\times 3\times 2\times 1)+(2\times 1)+1$
= 99
So, word “VARUN” is next after 99 words,
Then, the rank of the word “VARUN” is 100.

So, the correct answer is “Option (c)”.

Note: To solve such a type of question, always remember that we arrange the words here, so we use the concept of factorial denoted as n! and is evaluated as $n!=n(n-1)(n-2).....3.2.1$. Try not to leave any of the possible orders which occur before the arrangement of the word “VARUN” as this will reduce the rank of “VARUN”. Try not to make any calculation errors while solving the question.