How it works
To get from one corner to another using only moves right (R) and up (U), every shortest path uses the same number of each move. A path with r rights and u ups is a string like RRURU, so the number of paths is the number of ways to choose where the R's go:
C(r + u, r)
If the path must pass through a certain point, count the paths to that point and the paths from it, then multiply. If it must avoid a point, subtract the paths that go through it from the total. For small or irregular grids you can also write the number of ways to reach each corner, which is the sum of the numbers below it and to its left (this builds Pascal's triangle).
Grid Paths practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
You need 4 moves right and 4 moves up. How many shortest paths exist?
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Answer: 70
8 moves in total; choose which 4 are rights: C(8, 4) = 70.
Ant walks from (0,0) to (6,3) moving only right or up. Paths?
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Answer: 84
6 rights and 3 ups make 9 moves: C(9, 3) = 84.
You need 4 moves right and 3 moves up. Paths must pass through the point after 2 rights and 1 up. How many such paths exist?
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Answer: 18
To the point: 2 R and 1 U, C(3, 1) = 3 paths. From the point: 2 R and 2 U, C(4, 2) = 6 paths. Multiply: 3 × 6 = 18.
You need 6 moves right and 4 moves up. How many shortest paths exist?
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Answer: 210
10 moves; choose the 4 ups: C(10, 4) = 10 × 9 × 8 × 7 ÷ 24 = 210.
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Related topics
Related practice sets in the app
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