2016 is a nice binary number

2006 is the largest 11-digit binary number you can make with six 1's (and so five 0's). Therefore, any unique arrangement of 11111100000 will result in a unique number that is less than 2006. There are $\frac{11!}{6!\cdot5!}=462$ arrangements. We subtract 1 for 11111100000 itself (we are looking for numbers strictly less than 2006), so there are $462-1=\boxed{461}$ numbers.

As 2016 is having all the six 1's packed in left, shifting of any number of 1's to right results in a smaller number. All numbers possible with six 1's and five 0's are $\dbinom{11}{5} = 462$ . Leaving number 2016 out the answer is 461.