Difference between revisions of "2013 AIME I Problems/Problem 6"
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==Problem 6== | ==Problem 6== | ||
Melinda has three empty boxes and <math>12</math> textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Melinda has three empty boxes and <math>12</math> textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
− | ==Solution== | + | ==Solution One== |
The total ways the textbooks can be arranged in the 3 boxes is <math>( \,_{12}C_{3} )\cdot( \,_{9}C_{4} )</math>, which is equivalent to <math>12\cdot11\cdot10\cdot7\cdot3</math>. If all of the math textbooks are put into the box that can hold 3 textbooks, there are <math> \,_{9}C_{4} </math> ways for the other textbooks to be arranged. If all of the math textbooks are put into the box that can hold 4 textbooks, there are <math>9</math> ways to choose the other book in that box, times <math> \,_{8}C_{3} </math> ways for the other books to be arranged. If all of the math textbooks are put into the box with the capability of holding 5 textbooks, there are <math> \,_{9}C_{2} </math> ways to choose the other 2 textbooks in that box, times <math> \,_{7}C_{3} </math> ways to arrange the other 7 textbooks. <math> \,_{9}C_{4} =9\cdot7\cdot2</math>, <math>9\cdot( \,_{8}C_{3} )=9\cdot7\cdot8</math>, and <math>( \,_{9}C_{2} )\cdot( \,_{7}C_{3} )=9\cdot7\cdot4\cdot5</math>, so the total number of ways the math textbooks can all be placed into the same box is <math>(9\cdot7)(2+8+(4\cdot5))</math>. So, the probability of this occurring is <math>\frac{(9\cdot7)(2+8+(4\cdot5))}{12\cdot11\cdot10\cdot7\cdot3}</math>. If the numerator and denominator are both divided by <math>9\cdot7</math>, we have <math>\frac{(2+8+(4\cdot5))}{4\cdot11\cdot10}</math>. Simplifying the numerator yields <math>\frac{30}{10\cdot4\cdot11}</math>, and dividing both numerator and denominator by 10 results in <math>\frac{3}{44}</math>. This fraction cannot be simplified any further, so <math>m=3</math> and <math>n=44</math>. Therefore, <math>m+n=3+44=\boxed{047}</math>. | The total ways the textbooks can be arranged in the 3 boxes is <math>( \,_{12}C_{3} )\cdot( \,_{9}C_{4} )</math>, which is equivalent to <math>12\cdot11\cdot10\cdot7\cdot3</math>. If all of the math textbooks are put into the box that can hold 3 textbooks, there are <math> \,_{9}C_{4} </math> ways for the other textbooks to be arranged. If all of the math textbooks are put into the box that can hold 4 textbooks, there are <math>9</math> ways to choose the other book in that box, times <math> \,_{8}C_{3} </math> ways for the other books to be arranged. If all of the math textbooks are put into the box with the capability of holding 5 textbooks, there are <math> \,_{9}C_{2} </math> ways to choose the other 2 textbooks in that box, times <math> \,_{7}C_{3} </math> ways to arrange the other 7 textbooks. <math> \,_{9}C_{4} =9\cdot7\cdot2</math>, <math>9\cdot( \,_{8}C_{3} )=9\cdot7\cdot8</math>, and <math>( \,_{9}C_{2} )\cdot( \,_{7}C_{3} )=9\cdot7\cdot4\cdot5</math>, so the total number of ways the math textbooks can all be placed into the same box is <math>(9\cdot7)(2+8+(4\cdot5))</math>. So, the probability of this occurring is <math>\frac{(9\cdot7)(2+8+(4\cdot5))}{12\cdot11\cdot10\cdot7\cdot3}</math>. If the numerator and denominator are both divided by <math>9\cdot7</math>, we have <math>\frac{(2+8+(4\cdot5))}{4\cdot11\cdot10}</math>. Simplifying the numerator yields <math>\frac{30}{10\cdot4\cdot11}</math>, and dividing both numerator and denominator by 10 results in <math>\frac{3}{44}</math>. This fraction cannot be simplified any further, so <math>m=3</math> and <math>n=44</math>. Therefore, <math>m+n=3+44=\boxed{047}</math>. | ||
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+ | ==Solution Two== | ||
+ | It is easy enough to cycle through all <math>\frac{12!}{5! 4! 3!}</math> possibilities. Get a very large piece of scratch to draw them all out. Then, count the number of possibilities in which the three math textbooks are in the same box. | ||
== See also == | == See also == | ||
{{AIME box|year=2013|n=I|before=Problem 5|num-a=7}} | {{AIME box|year=2013|n=I|before=Problem 5|num-a=7}} |
Revision as of 16:40, 18 March 2013
Problem 6
Melinda has three empty boxes and textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as , where and are relatively prime positive integers. Find .
Solution One
The total ways the textbooks can be arranged in the 3 boxes is , which is equivalent to . If all of the math textbooks are put into the box that can hold 3 textbooks, there are ways for the other textbooks to be arranged. If all of the math textbooks are put into the box that can hold 4 textbooks, there are ways to choose the other book in that box, times ways for the other books to be arranged. If all of the math textbooks are put into the box with the capability of holding 5 textbooks, there are ways to choose the other 2 textbooks in that box, times ways to arrange the other 7 textbooks. , , and , so the total number of ways the math textbooks can all be placed into the same box is . So, the probability of this occurring is . If the numerator and denominator are both divided by , we have . Simplifying the numerator yields , and dividing both numerator and denominator by 10 results in . This fraction cannot be simplified any further, so and . Therefore, .
Solution Two
It is easy enough to cycle through all possibilities. Get a very large piece of scratch to draw them all out. Then, count the number of possibilities in which the three math textbooks are in the same box.
See also
2013 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |