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

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.