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Lattice paths in 3D

MediumCombinatorics~4m

How many shortest lattice paths go from (0,0,0)(0,0,0) to (3,4,5)(3,4,5), moving one unit at a time in the +x+x, +y+y, or +z+z direction?

Show hints (2)+
  1. A path is an ordering of 33 x-steps, 44 y-steps, 55 z-steps (1212 total).
  2. Multinomial: 12!/(3! 4! 5!)12!/(3!\,4!\,5!).

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27720

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