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Question: If all the words formed from the letters of the word HORROR are arranged in the opposite order as th...

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

Explanation

Solution

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 ∣! n ‾\left| \\!{\underline {\, n \,}} \right. is defined as; ∣! n ‾\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. ∣! n ‾\left| \\!{\underline {\, n \,}} \right. Is also denoted byn!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→\toR_____
=5!2!2!= \dfrac{{5!}}{{2!2!}}
=5×4×3×2!2×1×2!= \dfrac{{5 \times 4 \times 3 \times 2!}}{{2 \times 1 \times 2!}}
=30
Here, 5!5! for 5 letters is absent and 2!2! and 2!2! for O and R repeat two times respectively.
Number of words beginning with O→\toO_____
=5!3!= \dfrac{{5!}}{{3!}}
=5×4×3!3!= \dfrac{{5 \times 4 \times 3!}}{{3!}} =20
Hear, 5!5! for 5 letters is absent and 3!3! for repeat R three times.
Number of words beginning with HR→\toHR____
=4!2!2!= \dfrac{{4!}}{{2!2!}}
=4×3×2!2!×2×1= \dfrac{{4 \times 3 \times 2!}}{{2! \times 2 \times 1}}
=6
Here, 4!4!for 4 letters is absent and2!2! and 2!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  = 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!=10! = 1. Permutation is arrangements of letters. The different groups or selection of number or object is a combination. Number of combinations is crn=n!r!(n−r)!\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.