# Directed Graphs, Vertices and Distinct Paths

6. Recall that R^3={(x,y,z):x,y,z(subset of R)}. Let G(V,E) be a directed graph, in which V= {(x,y,z)-(subset of R^3) :x,y,z(subset of R),-10<=x,y,z<=10}.

Suppose that for any vertex, v=(x,y,z)--[subset of V], the only edges originating at v are the ones joining v to (x+1,y,z),(x,y+1,z),(x,y,z+1) .

i.e. any path that originates at v , must begin by moving one unit horizontally to

(x+1,y,z) or one unit vertically to (x,y+1,z) or one unit across to (x,y,z +1) .

a) How many distinct paths exist between(-1,2,0) and (1,3,7) ?

b) How many distinct paths exist between(1,0,5) and (8,1,7) ?

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6. Recall that . Let G(V,E) be a directed graph, in which . Suppose that for any vertex, v=(x,y,z) , the only edges originating at v are the ones joining v to (x+1,y,z),(x,y+1,z),(x,y,z+1). (i.e. any path that originates at v must begin by moving one unit horizontally to (x+1,y,z) or one unit vertically to (x,y+1,z) or ...

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