Up or Right

Starting at the point P(x,y) on the coordinate plane, a pin can be moved either to point A(x+1,y) or to point B(x,y+1). If the pin starts at (0,0) and is moved to (4,4), what is the probability that it passed through (2,2)?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
Under the given constraint the pin can only move either one unit up or one unit right at a time. The following figure shows there is 1 path from (0,0) to (0,1) and 1 path from (0,0) to (1,0). We write a 1 next to the points to indicate the number of path leading to the them

Likewise, there is 1 path from (0,0) to each of the other points on the vertical and horizontal axes

The number of paths from (0,0) to (1,1)=1+1=2

The number of paths from (0,0) to (2,1)=2+1=3

If we keep working this way, we find that there are 70 paths from (0,0) to (4,4)

There are 6 paths from (0,0) to (2,2) and by the same token 6 paths from (2,2) to (4,4). The number of paths from (0,0) to (4,4) passing through (2,2) equals
6\times 6=36
Probability of pin passing through (2,2) on the way from (0,0) to (4,4)  equals
36/70=18/35

Answer: 18/35

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About mvtrinh

Retired high school math teacher.
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