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# Arrangement of digits | AIME I, 2012 | Question 5

Try this beautiful problem from the American Invitational Mathematics Examination, AIME, 2012 based on Arrangement of Digits. You may use sequential hints.

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 $${11 \choose 7} = {330}$$.

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