How many unit paths are there in R^3 from (0, 0, 0) to (n, n, n) that never pass

Question

How many unit paths are there in R^3 from (0, 0, 0) to (n, n, n) that never pass below the plane y = x? This means that for every point (a, b, c) on the path we have a lessthanorequalto b. b) What happens if instead we insist that the paths never pass above the plane y = z? This means that we replace the condition "a lessthanorequalto b" with "a lessthanorequalto c". Let t N. a) Find a formula for the number of unit paths from (0, 0) to (n, n) that never pass below the line y-x-t. b) What does your formula give for t greaterthanorequalto n?

Explanation / Answer

1(a)

We have to go from (0,0,0) to (n,n,n) so that path does not pass below the plane y=x. If (a,b,c) be any point on the path, then we have to make sure that a<=b. Since there are infinitely many unit paths above the plane y=x which will lead from (0,0,0) to (n,n,n), therefore, there can be infinitely many units path from (0,0,0) to (n,n,n) which do not pass below the plane y=x.

1(b)

Again, there are infinitely many unit paths from (0,0,0) to (n,n,n) which pass below the y=z therefore, there are infinitely many unit paths that never pass above the plane y=z.

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