Moreover if the sequence of x i s and y i s satisfies these inequalities then

Moreover if the sequence of x i s and y i s satisfies

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Moreover, if the sequence of x i ’s and y i ’s satisfies these inequalities, then the sequence of intersections previously defined indicates a valid path. There are ( 5 k ) ways of choosing the k -tuple ( x 2 , x 4 , . . . , x 2 k ) and ( 4 k ) ways of choosing the k -tuple ( x 1 , x 3 , . . . , x 2 k - 1 ) subject to the required inequalities. Similarly, there are ( 5 k ) ways of choosing the k -tuple ( y 2 , y 4 , . . . , y 2 k ) and because y 2 k - 1 could be equal to y 2 k +1 = 5 , there are ( 5 k ) ways of choosing the k -tuple ( y 1 , y 3 , . . . , y 2 k - 1 , y 2 k +1 ) . Therefore the total number of possible paths is 4 k =0 ( 5 k ) 3 ( 4 k ) = 1 3 · 1 + 5 3 · 4 + 10 3 · 6 + 10 3 · 4 + 5 3 · 1 = 1 + 500 + 6000 + 4000 + 125 = 10626 . Difficulty: Hard NCTM Standard: Geometry Standard for Grades 9–12: use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.
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