Determine the number of positive integer x where $$x\le 9,999,999$$ and the sum of the digits in x equals 31
How do you approach this question?
TEXTBOOK SOLUTION:
Let x be written in base 10. The answer to this problem is the number of nonnegative integer solutions to $$x_1+x_2+x_3+x_4+x_5+x_6+x_7 = 31,\text{ } 0\le x_i,\text{ } 1\le i\le7 \text{ but } x_j \gt 9$$
How does this make sense?, why are there 7 terms of x. This could be arbitrary large, no? Maybe a bad question? Or a bad solution?
We have to count the words of length $7$ over the alphabet $\{0,1,2,\ldots,9\}$ which sum to $31$. (Allowing leading zeros automatically takes care of the numbers having less than seven decimal places.) This number can be realized as coefficient of $x^{31}$ in the expansion of $$F(x):=(1+x+x^2+\ldots+x^9)^7\ ,$$ because expanding $F$ according to the distributive law produces $10^7$ terms with coefficient $1$, each of them coming from one of the mentioned words, and then these terms can be collected according to increasing powers of $x$. The binomial series gives $$F(x)=\left({1-x^{10}\over 1-x}\right)^7=\bigl(1-7x^{10}+21x^{20}-35x^{30}-\ldots\bigr)\sum_{k=0}^\infty {7+k-1\choose k} x^k\ .$$ Summing the coefficients of the resulting $x^{31}$-terms we obtain $$N={37\choose 31}-7{27\choose 21}+21{17\choose 11}-35{7\choose 1}=512\,365\ .$$
One approach is to use stars and bars with inclusion-exclusion
$x_1 \; to \; x_7$ is there because there can be maximum $7$ digits ($7$ "cells"), with $0 \le x_i \le 9$
Using the formula $\binom{n+k-1}{k-1},\; with\; n = 31, k = 7,$
and excluding invalid cases by forcibly putting $10$ in combos of one or more "cells", and reducing the sum to be obtained accordingly, we get
$\binom{37}6 - \binom71\binom{27}6 + \binom72\binom{17}6 - \binom73\binom76 = 512,365$
Consider the number of digits in x and partitions of 31 into that many numbers where the maximum value in a part is 9. For example, consider that any 3 digit number will have at most 27 for the sum of its digits and so thus the cases to consider are 4,5,6 and 7 digits. This would be my suggestion as well as considering what re-arranging can be done at times along with adding zeros for the 4,5, and 6 cases as one could take a number like 4,999 which does add up to 31 and add zeroes into the number that complicates this a bit.
Inclusion-exclusion feels like it could become messy here.
My first attempt would probably be dynamic programming, building tables of the number of sequences of $n$ digits with sum $k$ for smaller $n$ (and $0\le k\le 31$) first.
