17. Choose 3 different numbers from 1 to 300, such that the sum of these 3 numbers is exactly divisible by 3. How many ways are there to do this?
17. The remainder when divided by 3 is $0,1,2$, so the 300 numbers from 1 to 300 can be divided into 3 categories: $A=\{1,4,7, \cdots, 298\}, B=\{2,5,8, \cdots, 299\}, \quad C=\{3,6,9, \cdots, 300\}$.
The number of ways to pick three numbers whose sum is exactly divisible by 3 is $3 C_{100}^{3}+100^{3}=1485100$ (ways).