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 are there, whose sum of digits equals 7?

## Solution

Consider the equation

Number of non negative integer solutions to this equation is

The set of solutions is bijective with the number of numbers less that whose digit adds up to 7. Hence the answer is

## Key Idea:

*Bijection; Number of Non Negative Integer Solutions to an equation*

But we cannot have the first digit to be zero. So, x_1>0.

Therefore we should reserve at least 1 value to x_1.

Therefore x_1+x_2+x_3+……+x_8=6 then all the non negative solutions should be accepted.

Since 1 has already been assigned to x_1.

That is not really necessary…all you need is the sum to be 7, sum of 8 whole numbers. Even if the first digit is 0,it doesn’t matter, because there is no specification as to the minimum number of digits, it can be anything from 1 to 8.