We have 6 digit numbers 000000 to 999999, and we need to find all of the possible combinations of these 6 digit numbers that sum to 27.
000999, 999000, etc all work. I understand this will involve build up counting, so we'll need to use $n \choose k$ for this problem. I've tinkered around with it and realized the sum of a string with the same value digit follows the following pattern:
So to get the sum to 27, we’d have to subtract 1 from some of the digits in the 555555 string 3 times, giving us some combination of 555444. If we do +/– some values on each digit, we can make the same combinations of 27. I just don't know how I can use $n \choose k$ on this pattern, where I use 555444 and its combinations +/- different values on each digit.
I also found this post here, which shows the use of the stars and bars strategy for solving a similar problem. However, I'm confused about how stars and bars applies in this case, or what the summation of the $n \choose k$ setup would look like
If someone can guide me in the right direction, I would greatly appreciate it. Thank you!