Question

If the letters of the word “MIRROR” are arranged as in a dictionary, then the rank of the given word is
A) 23
B) 84
C) 49
D) 48

Answer 1

It is given in the question that If the letters of the word “MIRROR” are arranged as in a dictionary, then what is the rank of the given word?
Firstly, we will arrange the given word “MIRROR” according to the dictionary. Then after, we will use permutation and combination.
Finally, solving further we will get the answer.
Standard formula of permutation and combination,
${}^n{p_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$

Complete step by step solution:
It is given in the question that If the letters of the word “MIRROR” are arranged as in a dictionary, then what is the rank of the given word?
Now, arrange the given word “MIRROR” according to the dictionary.
Then, the sequence of the word “MIRROR” would be IMORRR.
Since, we have to find the rank of the word “MIRROR”.
The sequence according to the dictionary is:
I
Since, after the alphabet “I”, we have five more alphabet to place.
But we can place them in three different ways as the alphabet “R” is repeated three times, and the alphabet “O” and “M”.
Therefore, the total ways of arrangement for rest five alphabets is:
$\therefore \dfrac{{5!}}{{3!}} = \dfrac{{5 \times 4 \times 3!}}{{3!}}$
$\therefore \dfrac{{5!}}{{3!}} = 5 \times 4$
$\therefore \dfrac{{5!}}{{3!}} = 20$ (I)
After the alphabet “I” there comes the probability for the alphabet “M” the follows the alphabet “I” and “O”.
M I O
So, the remaining alphabet can be arranged in:
$\therefore \dfrac{{3!}}{{3!}} = 1$ (II)
Since, “M” and “I” are fixed at the two places, in the 3rd place we can add “R” which is followed by “O”.
Therefore, the sequence can be:
M I R O
Hence, the remaining two alphabets can be arranged in:
$\therefore \dfrac{{2!}}{{2!}} = 1$ (III)
M I R R O
Therefore, the number of ways in which remaining place can be arranged is:
$1! = 1$ (IV)
Now, add equation (I), (II), (III) and (IV), we get,
$= 20 + 1 + 1 + 1 \\\ =23$

Therefore, if the letters of the word “MIRROR” are arranged as in a dictionary, then the rank of the given word is 23.

Note:
Combination: A combination is a selection of items from a collection, such that the order of selection does not matter.
${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Permutation: A permutation of a set is an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its element. The word permutation also refers to the act or process of changing the linear order of an order set.
${}^n{p_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$