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Difference between revisions of "1993 AIME Problems/Problem 7"

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== Problem ==
 
== Problem ==
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Three numbers, <math>a_1\,</math>, <math>a_2\,</math>, <math>a_3\,</math>, are drawn randomly and without replacement from the set <math>\{1, 2, 3, \dots, 1000\}\,</math>.  Three other numbers, <math>b_1\,</math>, <math>b_2\,</math>, <math>b_3\,</math>, are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let <math>p\,</math> be the probability that, after a suitable rotation, a brick of dimensions <math>a_1 \times a_2 \times a_3\,</math> can be enclosed in a box of dimensions <math>b_1 \times b_2 \times b_3\,</math>, with the sides of the brick parallel to the sides of the box.  If <math>p\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
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Three numbers, <math>a_1\,</math>, <math>a_2\,</math>, <math>a_3\,</math>, are drawn randomly and without replacement from the [[set]] <math>\{1, 2, 3, \dots, 1000\}\,</math>.  Three other numbers, <math>b_1\,</math>, <math>b_2\,</math>, <math>b_3\,</math>, are then drawn randomly and without replacement from the remaining set of 997 numbers.  Let <math>p\,</math> be the [[probability]] that, after a suitable rotation, a brick of dimensions <math>a_1 \times a_2 \times a_3\,</math> can be enclosed in a box of dimensions <math>b_1 \times b_2 \times b_3\,</math>, with the sides of the brick [[parallel]] to the sides of the box.  If <math>p\,</math> is written as a [[fraction]] in lowest terms, what is the sum of the numerator and denominator?
  
 
== Solution ==
 
== Solution ==
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{{solution}}
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Call the six numbers selected <math>\displaystyle x_1 > x_2 > x_3 > x_4 > x_5 > x_6</math>. Clearly, <math>x_1</math> must be a dimension of the box, and <math>b_6</math> must be a dimension of the brick.
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*If <math>x_2</math> is a dimension of the box, then any of the other three remaining dimensions will work as a dimension of the box. That gives us <math>3</math> possibilities.
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*If <math>x_2</math> is not a dimension of the box but <math>x_3</math> is, then both remaining dimensions will work as a dimension of the box. That gives us <math>2</math> possibilities.
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*If <math>x_4</math> is a dimension of the box but <math>x_2,\ x_3</math> aren’t, there are no possibilities (same for <math>x_5</math>).
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The total number of arrangements is <math>{6\choose3} = 20</math>; therefore, <math>p = \frac{3 + 2}{20} = \frac{1}{4}</math>, and the answer is <math>1 + 4 = 005</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1993|num-b=6|num-a=8}}
 
{{AIME box|year=1993|num-b=6|num-a=8}}
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[[Category:Intermediate Combinatorics Problems]]

Revision as of 19:40, 27 March 2007

Problem

Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}\,$. Three other numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\,$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3\,$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3\,$, with the sides of the brick parallel to the sides of the box. If $p\,$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?

Solution

Call the six numbers selected $\displaystyle x_1 > x_2 > x_3 > x_4 > x_5 > x_6$. Clearly, $x_1$ must be a dimension of the box, and $b_6$ must be a dimension of the brick.

  • If $x_2$ is a dimension of the box, then any of the other three remaining dimensions will work as a dimension of the box. That gives us $3$ possibilities.
  • If $x_2$ is not a dimension of the box but $x_3$ is, then both remaining dimensions will work as a dimension of the box. That gives us $2$ possibilities.
  • If $x_4$ is a dimension of the box but $x_2,\ x_3$ aren’t, there are no possibilities (same for $x_5$).

The total number of arrangements is ${6\choose3} = 20$; therefore, $p = \frac{3 + 2}{20} = \frac{1}{4}$, and the answer is $1 + 4 = 005$.

See also

1993 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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