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Test of Mathematics Solution Subjective 50 -Dictionary Ranking

In Progress This is a Test of Mathematics Solution Subjective 50 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Problem

All the permutation of the letters $$a,b,c,d,e$$ are written down and arranged in alphabetical order as in dictionary. Thus the arrangement $$abcde$$ is in first position and $$abced$$ is in second position. What is the position of the word $$debac$$?

Solution:

According to the arrangement in a dictionary, number of words:

i) starting with $$a$$ = 4!

ii) starting with $$b$$ = 4!

iii) starting with $$c$$ = 4!

iv) starting with $$da$$ = 3!

v) starting with $$db$$ = 3!

vi) starting with $$dc$$ = 3!

vii) starting with $$dea$$ = 2!

viii) starting with $$deba$$ = 1!

The last case gives us the word itself. So the position of the word will be = $$3*4! + 3*3! + 2! +1$$ = $$93$$

$$debac$$ is in the $$93^{rd}$$ position.