How many different combinations are there of three distinct integers between one and thirty (inclusive) such that their sum is a multiple of three

Question

Now, there are ten numbers of each of the following groups; we shall name one, two and three: $3x, 3x+1$ or $3x+2$.

So you can have 3 of group one, 3 of group two, or one of each from each of the groups. This gives ${10 \choose 3} + {10 \choose 1}^3 + {10 \choose 3} = 1240$.

However, the answers suggest that there is an extra $10C3$ to be added on (making $1360$). Where does this come from?

Answer 1

You can have 3 of Group 3: $2+2+2 \equiv 0 \pmod{3}$.