Question

If all the words formed from the letters of the word HORROR are arranged in the opposite order as they are in a dictionary, then the rank the rank of the words HORROR is
A. 56
B. 57
C. 58
D. 59

Answer 1

Hint –In this question, this type of question comes in permutation and combination. This question belongs to permutation. If n be a positive integer then factorial n, denoted $\left| \!{\underline {\,
  n \,}} \right. $is defined as; $\left| \!{\underline {\,
  n \,}} \right. $=n (n-1) (n-2)....3.2.1 .if same type of letter repeated then no of repeat letters is in fraction, which is shown in below. $\left| \!{\underline {\,
  n \,}} \right. $Is also denoted by$n!$.

Complete step-by-step answer:
   In this question, the words is given - H, R, O
The correct arrangements is H, O, R and
The opposite arrangements is R, O, H
First of all we see
Number of words beginning with R$ \to $R_____
                                     \[ = \dfrac{{5!}}{{2!2!}}\]
                                    \[ = \dfrac{{5 \times 4 \times 3 \times 2!}}{{2 \times 1 \times 2!}}\]
                                     =30
Here, $5!$ for 5 letters is absent and \[2!\] and \[2!\] for O and R repeat two times respectively.
Number of words beginning with O$ \to $O_____
                                       $ = \dfrac{{5!}}{{3!}}$
                                      $ = \dfrac{{5 \times 4 \times 3!}}{{3!}}$ =20
 Hear, $5!$ for 5 letters is absent and $3!$ for repeat R three times.
Number of words beginning with HR$ \to $HR____
                                            $ = \dfrac{{4!}}{{2!2!}}$
                                            $ = \dfrac{{4 \times 3 \times 2!}}{{2! \times 2 \times 1}}$
                                                =6
Here, $4!$for 4 letters is absent and$2!$ and $2!$ for O and R repeat two times respectively.
  The following words are HORRRO AND HORROR.
   TOTAL NUMBER OF WORDS IS
                                                    $
   = 30 + 20 + 6 + 1 + 1 \\
   = 58 \\
 $
  The rank of the given word HORROR is 58. So option C is correct.

Note – We define the value of $0! = 1$. Permutation is arrangements of letters. The different groups or selection of number or object is a combination. Number of combinations is $\mathop c\limits_r^n = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ where, the number of all combinations of things, taken r times. A combination is an arrangement of items in which order does not matter but a permutation is an arrangement of items in a particular order which order matters.