Question

Words with meaning or without meaning are formed in the order opposite to that of in a dictionary, then the rank of the word 'HORROR' is \[\]
A.57 \[\]
B.58 \[\]
C.59 \[\]
D.56 \[\]

Answer 1

Hint: We see that in opposite to dictionary order the letters will be in the following sequence R, O, H. We find the number of words by beginning with R, beginning with O, and beginning with HR. We add the number of words and add 2 because to get the rank because the next words that will be read are HORRRO and HORROR.

Complete step-by-step solution:
‘We know that if there are $n$ objects with $m$ objects repeat themselves ${{p}_{1}},{{p}_{2}},...,{{p}_{m}}$ times then the number of arraignments is given by $\dfrac{n!}{{{p}_{1}}!{{p}_{2}}!...{{p}_{m}}!}$ \[\]
We are given the word ‘HORROR’ where there are 6 letters with H occurring once. O repeating itself 2 times and R repeating itself 3 times. We are asked to find the rank of the word ‘HORROR’ in reverse dictionary order. We see that in reverse dictionary order the letters will be in the following sequence: R, O, H.
\[\begin{matrix}
   R & \_ & \_ & \_ & \_ & \_ \\
\end{matrix}\]
Let us find the number of words beginning with R. We fix one R at the first position and arrange the rest $6-1=5$ letters where O repeats itself 2 times and R repeats itself 2 times in $\dfrac{5!}{2!2!}=30$ ways. So the number of words beginning with R is 30.
\[\begin{matrix}
   O & \_ & \_ & \_ & \_ & \_ \\
\end{matrix}\]
Let us find the number of words beginning with O. We fix one O at the first position and arrange the rest 5 letters where R repeats itself 3 times in $\dfrac{5!}{3!}=20$ ways. So the number of words beginning with O is 20.
\[\begin{matrix}
   H & R & \_ & \_ & \_ & \_ \\
\end{matrix}\]
Now following the reverse dictionary order afterward beginning with R and O are the words beginning with H will appear. We fix H in the first position and R in the second position and find the number of words beginning with HR by filling the rest $6-1-1=4$ positions with 4 letters where R repeat itself 2 times and O repeat itself 2 times as $\dfrac{4!}{2!2!}=6$ \[\]
 We see that after the words beginning with HR, the words with HO will appear in the reverse dictionary. The words are HORRRO and then HORROR.
So the rank of the word horror in reverse dictionary order is $30+20+6+2=58$. \[\]

Note: We must be careful that we are asked the rank in reverse dictionary order not in dictionary order. The rank of a word in dictionary order is the position of the word when the letters of the word are arranged to form different words in alphabetical order with or without meaning. We note that if there are no repetitions we can arrange $n$ objects in $r$ places in ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$ way.