I saw an example of such a problem here already, where the sum of digits desired was 8. I have already done my question using multinomial theorem, by finding out the coefficient of x^18 in $(x^0+x^1+...+x^9)^6$, and the proper answer came (which is 25927). How to solve it using the fact that $x_1+x_2+...+x_6=18$, and hence through $18+5\choose 5$? I observed the fact that I get the term $18+5\choose 5$ in the first way also, but some options get eliminated after that (subtracted). I understand that those are the cases where the digits are something greater than 9, but is there a proper mathematical explanation for it? Thanks in advance.
Well, I think you gave the explanation yourself already: the $18+5\choose 5$ is for any $6$ (ordered!) numbers that add up to $18$, and that includes more cases than having $6$ digits that add up to $18$ .. I am not sure what else you might be looking for in order to explain this.
But maybe this helps ....
In terms of stars and bars: finding the ways in which $6$ numbers add up to $18$, you'd need to put $5$ bars between or on the sides of $18$ stars, e.g
corresponds to the ordering of $6$ numbers $(4,2,0,1,11,0)$, and you can see there are $18+5\choose 5$ ways to put those $5$ bars in those $18+5=23$ positions
However, when it comes to $6$-digit numbers (where a number like $53$ is represented by $000053$), you can't put the bars more than $9$ spots apart, and so there are fewer ways to do that.