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

## Check the Answer

But try the problem first…

Answer: is 330.

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 \({11 \choose 7} = {330}\).

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s

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