Solution
Paths Below the Diagonal
Show the problem again
How many lattice paths from (0, 0) to (10, 10) using only unit right and up steps never rise above the diagonal y = x?
Worked solution
The answer is 16,796, the 10th Catalan number C₁₀ = C(20, 10)/11. The reflection principle proves it: of the C(20, 10) unrestricted paths, the bad ones (touching y = x + 1) biject with paths to (9, 11) via reflecting the prefix, giving C(20,10) − C(20,9) = C(20,10)/11 = 184756/11 = 16796.
Source: Catalan-number lattice-path problem; the reflection argument is due to Desire Andre (1887). Statement written for AxiomIQ.