2012
Problem - 214
Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.
When $1$ is subtracted from a binary number, the number of digits will remain constant if and only if number to be subtracted ended in $10$. Therefore, every subtraction involving two numbers from $B$ will necessarily involve exactly one number ending in $10$.
Therefore, the answer will be the same as the number of numbers ending with $01$. With the $10$ in place, the seven remaining $1$'s can be put in any of the remaining $11$ spaces, so the answer is ${11 \choose 7} = \boxed{330}$.