Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution Subjective 60 (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

Consider the set S of all integers between and including 1000 and 99999. Call two integers x and y in S to be in the same equivalence class if the digits appearing in x and y are the same. For example, if x = 1010, y = 1000 and z = 1201, then x and y are in the same equivalence class, but y and z are not. Find the number of distinct equivalence classes that can be formed out of S.

Solution

Any set of distinct digits with maximum order 5 is a equivalence class that can be formed out of S except {0}.
Number of such sets is
= {\dbinom {10}{1}} + {\dbinom {10}{2}} + {\dbinom {10}{3}} + {\dbinom {10}{4}} + {\dbinom {10}{5}} -1
= 10 + 45 + 120 + 210 + 252 -1
= 636