Question 1196433
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The meaning of the word  " between "  in  Math problems is not precisely defined.

Every time you should explain,  if endpoints are included or not.

So,  I will re-formulate the problem in this form


<pre>
    +---------------------------------------------------------------------+
    |   How many numbers from 10 to 999 inclusive have a digit sum of 8?  |
    +---------------------------------------------------------------------+
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Solution</U>


<pre>
For a minute, I will change my problem and ask

    +-----------------------------------------------------------------------+
    |   How many numbers from 0 to 999 inclusive have a digit sum of 8?    |
    +-----------------------------------------------------------------------+


Then it is the same as to ask

   
    How many integer solutions does this equation have 

        {{{x[1]}}} + {{{x[2]}}} + {{{x[3]}}} = 8

    in integer non-negative numbers  {{{x[1]}}} >= 0,  {{{x[2]}}} >= 0,  {{{x[3]}}} >= 0 ?


The answer is known from Combinatorics ( " Stars and Bars method " ),

( see this Wikipedia article  

https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)#:~:text=In%20the%20context%20of%20combinatorial,his%20classic%20book%20on%20probability. )


The number of solution in this case is  {{{C[8+3-1]^(3-1)}}} = {{{C[10]^2}}} = {{{(10*9)/2}}} = 45.


From it, we should subtract 1, which represents the number "8",
(we included it, when extended the set from [10,999] to [0,999]).


After subtracting, we get the final 


<U>ANSWER</U>.  There are 45-1 = 44  integer numbers from 10 to 999 (inclusive), such that the sum of their digits is 8.
</pre>

Solved.