Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 57 (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

How many 6-letter words can be formed using the letters A,B and C so that each letter appears at least once in the word.

Solution:

Number of ways 6 letter words can be formed using letters A,B,C so that each letter appears at least once in the word = Number of ways 6 letter word formed using A,B,C – Number of ways 6 letter word formed using any two of A,B,C + Number of 6 letter words formed by using A,B,C. [ using inclusion exclusion principle. ]

= {3^6} {\binom{3}{2}}{2^6} + {\binom{3}{1}}{1^6}

= 555 [ans]