Problem
COM-B1-M12-P023 Set 6. Diagonal
#23
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How many paths from \((0,0)\) to \((4,4)\), using right and up moves, never go above the diagonal \(y=x\)?
Subtract bad paths by reflection.
There are \(\binom{8}{4}=70\) paths total. Bad paths that first go above the diagonal reflect to paths from \((-1,1)\) to \((4,4)\), of which there are \(\binom{8}{3}=56\). Therefore the number of good paths is \(70-56=14\).
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