Test of Mathematics at the 10+2 Level

This is a Test of Mathematics Solution (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 natural numbers less that 10^8 are there, whose sum of digits equals 7?

Solution

Consider the equation x_1 + x_2 + ... + x_8 = 7

Number of non negative integer solutions to this equation is \dbinom{14}{7}

The set of solutions is bijective with the number of numbers less that 10^8 whose digit adds up to 7. Hence the answer is \dbinom{14}{7}

Key Idea:

Bijection; Number of Non Negative Integer Solutions to an equation