2. How many ways are there to insert + 's between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30 ?
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 $\binom{14}{9}=2002$.