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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 word ANIMAL is x, then the
permutation with rank x. among the permutations obtained by permuting the letters of the
word PERSON and arranging the permutations thus formed in dictionary order is
Answers
An easy question
Look
Basically I am not going to do it completely
Just giving you an idea
ANIMAL can have total arrangements = 6!/2! as two A are similar = 360
ANIMAL = AAILMN
(alphabetical order)
Number of words before ANIMAL (think as you are opening up the dictionary made up of these 6 letters __ assume there are only 5 alphabets AILMN and not 26 in ENGLISH _ question will become easy _ Words with A _ then Words with I _ then words with L _ then words with M _ then words with N _ Dictionary over)
=
WORDs starting with A in Dictionary are 0(nothing before A in AAILMN)
+
WORDs starting with AN in Dictionary __ 4C1 (AILM before N in _AILMN as A is already used) * 4! (to arrange at last four places after AN in ANIMAL)
+
WORD starting with ANI in Dictionary __ 1C1 (A before I in _AILM_ as A and N are already used) * 3! (to arrange at last three places after ANI in ANIMAL)
+
WORD starting with ANIM in Dictionary __ 2C1 (A and L before M in _A_LM_ as A,I and N are already used) * 2! (to arrange at last two places after ANIM in ANIMAL)
+
WORD starting with ANIMA in Dictionary __ 0 (nothing before A in _A_L_._ as A,I,M and N are already used) * 1! (to arrange at last place after ANIMA in ANIMAL)
= 0*5! + 4*4! + 1*3! + 2*2! + 0*1!
= 106
Clearly word after ANIMA in last process above is ANIMAL
RANK of ANIMAL = 107th
So x = 107
Similarly for PERSON
PERSON can have total arrangements = 6! = 720
So there will be (120 + 120 + 120 + 120 + 120 + 120 = 720) words
There will be 120 Words starting with E , N , O , P , R and S each
Clearly we need to find 107th word in this NEW Dictionary
PERSON = ENOPRS
(alphabetical order)
(think as you are opening up the dictionary made up of these 6 letters __ assume there are only 6 alphabets ENOPRS and not 26 in ENGLISH _ question will become easy _ Words with E _ then Words with N _ then Words with O _ then Words with P _ then Words with R _ then Words with A _ Dictionary over)
Total words with E (first letter in new dictionary ENOPRS) = 120
WORDs starting with E in Dictionary __ 1 (nothing before E in ENOPRS ) * 5! (to arrange at last five places after E in ENOPRS ) = 120
We need 107th word
So
Words starting with EN , EO , EP, ER , ES each = 24
then only we will have 5 ( EN , EO , EP, ER , ES ) * 24 = 120
So if we will subtract all words starting from ES (last in the above 5 patterns of EN , EO , EP, ER , ES ) from total words starting from E then
= 120 - 24 = 96
This means 107th word starts from ES confirmed
Words starting with ESN , ESO , ESP , ESR each = 6
then only we will have 4 ( ESN , ESO , ESP, ESR ) * 6 = 24 __total words starting from ES
So if we will subtract all words starting from ESR and ESP (last in the above 4 patterns of ESN , ESO , ESP, ESR ) from total words starting from E then
= 120 - (6 for starting with ESR) - (6 for starting with ESP)
= 108
Clearly 107th word not starts from ESR and ESP
but from ESO
All words from ESO =
ESONPR (103rd word)
ESONRP (104th word)
ESOPNR (105th word)
ESOPRN (106th word)
ESORNP (107th word)
ESORPN (108th word)
Answer is ESORNP (107th word)
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