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| − | == Problem ==
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| − | Objects <math>A</math> and <math>B</math> move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object <math>A</math> starts at <math>(0,0)</math> and each of its steps is either right or up, both equally likely. Object <math>B</math> starts at <math>(5,7)</math> and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?
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| − | <math>
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| − | \mathrm{(A)} \ 0.10 \qquad
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| − | \mathrm{(B)} \ 0.15 \qquad
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| − | \mathrm{(C)} \ 0.20 \qquad
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| − | \mathrm{(D)} \ 0.25 \qquad
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| − | \mathrm{(E)} \ 0.30 \qquad
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| − | </math>
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| − | == Solution ==
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| − | If <math>A</math> and <math>B</math> meet, their paths connect <math>(0,0)</math> and <math>(5,7).</math> There are <math>\binom{12}{5}=792</math> such paths, so the probability is <math>\frac{792}{2^{6}\cdot 2^{6}} \approx 0.20 \Rightarrow \boxed{C}</math>
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| | == See Also == | | == See Also == |
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| | {{AMC12 box|year=2003|ab=A|num-b=23|after=25}} | | {{AMC12 box|year=2003|ab=A|num-b=23|after=25}} |
