2012 AIME I Problems/Problem 5
Let be the set of all binary integers that can be written using exactly zeros and ones where leading zeros are allowed. If all possible subtractions are performed in which one element of is subtracted from another, find the number of times the answer is obtained.
When is subtracted from a binary number, the number of digits will remain constant if and only if the original number ended in Therefore, every subtraction involving two numbers from will necessarily involve exactly one number ending in To solve the problem, then, we can simply count the instances of such numbers. With the in place, the seven remaining 's can be distributed in any of the remaining spaces, so the answer is .
https://www.youtube.com/watch?v=f5ZoAFfc-1E&list=PLyhPcpM8aMvIo_foUDwmXnQClMHngjGto&index=5 (Solution by Richard Rusczyk) - AMBRIGGS
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