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.