, 2017

Solution:

Answer: $2002$

Note that because there are $15$ 1's, no matter how we insert +'s, the result will always be a multiple of $3$. Therefore, it suffices to consider adding +'s to get a multiple of $10$. By looking at the units digit, we need the number of summands to be a multiple of $10$. Because there are only $15$ digits in our number, we have to have exactly $10$ summands. Therefore, we need to insert $9$ +'s in $14$ possible positions, giving an answer of $\binom{14}{9} = 2002$.