Question

If the letters of the word “MIRROR” are arranged as in a dictionary, then the rank of the given word is
$
  A.{\text{ 23}} \\
  B.{\text{ 84}} \\
  {\text{C}}{\text{. 49}} \\
  {\text{D}}{\text{. 48}} \\
 $

Answer 1

Hint: Here we are going to use combination and permutations standard formula. $
  ^nc{}_r = \dfrac{{n!}}{{r!(n - r)!}} \\
  {}^np{}_r = \dfrac{{n!}}{{(n - r)!}} \\
 $

Complete step by step answer:
Arrange the given word “MIRROR” according to the dictionary.
The sequence would be – IMORRR
Here we have to find the rank of “MIRROR”
The sequence according to the dictionary is-
$I\,\underline {} {\text{ }}\underline {} \;\underline {} {\text{ }}\underline {} \;\underline {} $
After the alphabet “I” we have five more alphabets to place. But we can place them in three different ways as the alphabet “R” is repeated three times, and the alphabets O and M.
Therefore the total ways of arrangement for rest five alphabets is –
$
  \dfrac{{5!}}{{3!}} \\
   = \dfrac{{5 \times 4 \times 3!}}{{3!}} \\
   = 5 \times 4 \\
   = 20\;{\text{ }}.........{\text{(A)}} \\
 $ (Simplify the equation)
After the alphabet “I” there comes the probability for the alphabet “M” then follows the alphabets “I” and “O” .
$M{\text{ I O}}\underline {} {\text{ }}\underline {} {\text{ }}\underline {} $
So, here remaining alphabets can be arranged in –
$\dfrac{{3!}}{{3!}} = 1$ ........... (B)
Now ‘M’ and ‘I’ are fixed at the 2 places, in the 3rd place we can add ‘R’ which is followed by ‘O’, Therefore the sequence can be –
$M{\text{ I R O }}\underline {{\text{ }}} \;\underline {} $
Hence remaining two alphabets can be arranged in –
$\dfrac{{2!}}{{2!}} = 1$ ......... (C)
$M{\text{ I R R O }}\underline {} $ The number of ways in which remaining place can be arranged is
$1! = 1$ .......... (D)
Adding equations (A), (B), (C) and (D)
\[
   = 20 + 1 + 1 + 1 \\
   = 23 \\
 \]
Therefore, if the letters of the word “MIRROR” are arranged as in a dictionary, then the rank of the given word is $23$

Hence from given multiple choices, option A is the correct answer.

Note: Combinations are used if the certain objects are to be arranged in such a way that the order of objects is not important whereas the permutation is an ordered combination- an act of arranging the objects or numbers in the specific or the favourable order.