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# Test of Mathematics Solution Subjective 7 – Sum of the Digits

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.

Also visit: I.S.I. & C.M.I. Entrance Course of Cheenta

## 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