How many positive integers, less than 2011, are multiples of 3 or 4, but not of 5?
For any integer $k$, there are $\lfloor 2011 / k\rfloor$ multiples of $k$ less than 2011. Therefore, there are 670 multiples of 3, and 502 multiples of 4. To count the numbers that are multiples of 3 or 4, we need to count the multiples of 3, the multiples of 4, and subtract the numbers that are multiples of both 3 and 4, which are counted twice. These numbers are the multiples of 12, and there are 167 of them. Therefore, there are $670+502-167=1005$ numbers less than 2011 that are multiples of 3 or 4.

Now, we need to subtract from this count the multiples of 5. A number is a multiple of 3 and 5 if and only if it is a multiple of 15: there are 134 of them, and a number is a multiple of 4 and 5 if and only if it is a multiple of 20: there are 100 of them. We need to subtract from this count the numbers that are multiples of 3, 4, and 5, which are counted twice. These are the multiples of 60, and there are 33 of them. Finally, the number sought is $1005-(134+100-33)=804$.