Difference between revisions of "2012 AIME I Problems/Problem 5"
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− | == Problem | + | == Problem == |
Let <math>B</math> be the set of all binary integers that can be written using exactly <math>5</math> zeros and <math>8</math> ones where leading zeros are allowed. If all possible subtractions are performed in which one element of <math>B</math> is subtracted from another, find the number of times the answer <math>1</math> is obtained. | Let <math>B</math> be the set of all binary integers that can be written using exactly <math>5</math> zeros and <math>8</math> ones where leading zeros are allowed. If all possible subtractions are performed in which one element of <math>B</math> is subtracted from another, find the number of times the answer <math>1</math> is obtained. | ||
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When <math>1</math> is subtracted from a binary number, the number of digits will remain constant if and only if the original number ended in <math>10.</math> Therefore, every subtraction involving two numbers from <math>B</math> will necessarily involve exactly one number ending in <math>10.</math> To solve the problem, then, we can simply count the instances of such numbers. With the <math>10</math> in place, the seven remaining <math>1</math>'s can be distributed in any of the remaining <math>11</math> spaces, so the answer is <math>{11 \choose 7} = \boxed{330}</math>. | When <math>1</math> is subtracted from a binary number, the number of digits will remain constant if and only if the original number ended in <math>10.</math> Therefore, every subtraction involving two numbers from <math>B</math> will necessarily involve exactly one number ending in <math>10.</math> To solve the problem, then, we can simply count the instances of such numbers. With the <math>10</math> in place, the seven remaining <math>1</math>'s can be distributed in any of the remaining <math>11</math> spaces, so the answer is <math>{11 \choose 7} = \boxed{330}</math>. | ||
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+ | ==Video Solutions== | ||
+ | |||
+ | https://www.youtube.com/watch?v=cQmmkfZvPgU&t=30s | ||
+ | |||
+ | https://www.youtube.com/watch?v=f5ZoAFfc-1E&list=PLyhPcpM8aMvIo_foUDwmXnQClMHngjGto&index=5 (Solution by Richard Rusczyk) - AMBRIGGS | ||
== See also == | == See also == | ||
{{AIME box|year=2012|n=I|num-b=4|num-a=6}} | {{AIME box|year=2012|n=I|num-b=4|num-a=6}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 17:14, 30 July 2022
Contents
Problem
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.
Solution
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
.
Video Solutions
https://www.youtube.com/watch?v=cQmmkfZvPgU&t=30s
https://www.youtube.com/watch?v=f5ZoAFfc-1E&list=PLyhPcpM8aMvIo_foUDwmXnQClMHngjGto&index=5 (Solution by Richard Rusczyk) - AMBRIGGS
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.