Question

All the letters of the word ANIMAL are permuted in all possible ways and the permutations thus formed are arranged in dictionary order. If the rank of the ANIMAL is x, then the permutations with rank x, among the permutations obtained by permuting the letter of the word PERSON and arranging the permutations thus formed in dictionary order is

Answer 1

Hint: To find the rank of the word ANIMAL in the permutations of letters of the word ANIMAL, we have to fix the letters in alphabetical order step by step. Once we find the rank of the word ANIMAL, we will permute the letters of word PERSON by fixing the letters in alphabetical orders, until we get to the rank x.

Complete step-by-step answer:
Alphabetically, the first letter from the letters of the word ANIMAL is A. But there are two A’s in the word ANIMAL.
Thus, we will fix the first letter as in the word ANIMAL we have A in the first place and find all the words with the second place of the word as A. This will leave 4 places which the other letters can occupy in any order.
A A _ _ _ _
These 4 places, thus can be occupied in 4! ways = 24 ways.
Then after this, we will get the series of words in which the first letter is A and the second letter is I, which is next in the alphabetical order.
A I _ _ _ _
Again, 4 places are left and can be occupied in 4! Ways = 24 ways.
Then, the next letter in alphabetical order is L.
A L _ _ _ _
There will be 4! words = 24 words.
After this, M will occupy second place and we will find all the words that start with AM.
A M _ _ _ _
These will be 4! = 24 words.
Now, next in order is N, and we need N in second place for ANIMAL. Thus, we will fix AN and find words starting with ANA,
A N A _ _ _
Three places are left and thus we will have 3! = 6 words.
Next in order is I, and we want I in the third place for animals. So, we will now fix the first three places for ANI and find all the words with A in the fourth place.
A N I A _ _
2 places are left, so only 2 words will be formed.
Next will be L in the fourth place.
A N I L _ _
2 places, so 2 words.
Next in order is M, and we need M in fourth place. For fifth place we will have A and we need it at that place. The only letter left now will N and we need it in last place and thus we find the word ANIMAL.
Therefore, the rank x is (24 + 24 + 24 + 24 + 6 + 2 + 2 + 1) = 107
Now, we will permute the letters of the word PERSON and find the word at the rank 107 if all the words are arranged in alphabetical order.
First of all we will try with first letter in alphabetical order in PERSON, I,e, E
E _ _ _ _ _
There are 5 spaces and so the words that start with E are 5! = 120. But this is more than 107. So, we know that the first letter is definitely E.
We will fix E and try for the second letter in alphabetical order, i.e E again.
E E _ _ _ _
There are 4 spaces, so 4! = 24 words.
Similarly, we will try with O P R.
Thus, starting with EO will be 4! = 24
Words starting with EP will be 4! = 24
Words starting with ER will be 4! = 24
Total rank of the last word starting with ER will be (24 + 24 + 24 + 24) = 96
Thus, the word will definitely start with ES as starting letters. So, we will fix ES and try for N as the third letter.
E S N _ _ _
3 spaces, so, 3! = 6 words.
Now the rank of the last word with ESN will be 96 + 6 = 102. Now, we definitely know that the third letter is O as if we permute with ESO, there will be 6 permutations and rank will be 108.
Next letter is N.
E S O N _ _
2 places, so 2 words. Total rank = 104
E S O P _ _
2 places, so 2 words. Total rank = 106
E S O R N P.
Thus, the rank of ESORNP has the rank 107.

Note: n! is defined as n(n – 1)(n – 2)…1. While solving such problems of rank, students are advised to write the letters of the word alphabetically as sometimes it gets confusing which letters are already used and which are yet to be used. One can notice that the word ANIMAL and PERSON have the same attributes. Both are 6 letter words and their first alphabetical letter is twice in the word. Thus, we can observe similar patterns in their permutations for the same rank, which can be used by students to skip some steps.