How many $4$ digit numbers have a digit sum of $11$?

Question

I used stars and bars for this problem, where there are $13$ spots and $3$ bars. This is $13$ choose $3$, but there are overcounts when $10$ and $11$ are a "digit." There are $4$ possibilities when $11$ is the first "digit" or $10$ as the last $3$ digits. there are $3$ choose $2$ possibilities. This is $286-7=279$. Am I correct?

Answer 1

The baseline of fourteen stars giving thirteen spots for bars is correct. I agree with your answer for the overcount, but do not understand the reasoning behind it. Having an $11$ is an overcount. That can only come from $1+1+1+11$ in any order, so we deduct $4$. Having a $10$ in one of the last three groups is not an overcount because that represents a digit of $9$. It is only when the first group is $10$ that you have an overcount. The last three groups can then be $1+1+2$ in some order, so you need to subtract three.