Difference between revisions of "2012 AIME I Problems/Problem 5"
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== Solution == | == Solution == | ||
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| + | When a number in binary notation is subtracted by 1, it will contain the same number of digits as the original only if it originally ended in the digits "10". Therefore all the binary numbers that fit the conditions of this problem end in the digits "10". All the other 7 1's can be distributed in the remaining 11 spaces, and so the answer is <math>{11\choose 7}= \boxed{330}</math>. | ||
== 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}} | ||
Revision as of 00:08, 17 March 2012
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.
Solution
When a number in binary notation is subtracted by 1, it will contain the same number of digits as the original only if it originally ended in the digits "10". Therefore all the binary numbers that fit the conditions of this problem end in the digits "10". All the other 7 1's can be distributed in the remaining 11 spaces, and so the answer is 
.
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 | ||
