 Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2012 based on Arrangement of digits.

## Arrangement of digits – AIME 2012

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.

• is 107
• is 330
• is 840
• cannot be determined from the given information

### Key Concepts

Arrangements

Algebra

Number Theory

But try the problem first…

Source

AIME, 2012, Question 5

Combinatorics by Brualdi

## Try with Hints

First hint

When 1 subtracts from a number, the number of digits remain constant when the initial number has units and tens place in 10

Second Hint

Then for subtraction from B requires one number with unit and tens place 10.

Final Step

10 there, remaining 1 distribute any of other 11 then answer ${11 \choose 7} = {330}$.