For how many integers from 1 through 99,999 is the sum of their digits equal to 10?

To Find :

The number of integers from 1 through 99,999 which the sum of their

digits equal to 10.

Given information:

The integers from 1 up to 99,999 and consider their digits that sum up to 10.

Formulae used:

The number of non-negative solutions of the equation is equal to

( n+k−1n) =(n+k−1)!n!(k−1)!.

Calculation:

First consider two-digit numbers among the given set of integers that sum up to 10.

Hence, for x1+x2 =10

There are;

=( 10+2−1 10 )=( 11 10 )=11!1!( 10)!=11

There are 11 solutions.

But from 11 solutions we must subtract (0,10) and (10,0).

Then so we have 9 solutions such as 19,91,28,82,37,73,46,64,55.

We need to consider the given full integer set from 1 through 99,999.

So, as we first determined the two digits that sum up to 10 and found the solutions.

Next we must determine three digits that sum up to 10 and so on.

Therefore three digits that sum up to 10

That is;

For x1+x2+x3=10

The solutions are;

 =( 10+3−1 10 )= ( 12 10 ) = 12!2!( 10)!= (12*11) / 2 =66

63 solutions to be considered since we have to discard the solutions (10,0,0) ,(0,10,0),(0,0,10)