How Many Paths Exist from (0, 0, 0) to (a, b, c) in a 3D Grid?

[dancergirlie]

Homework Statement

Let a, b, and c be positive integers. How many paths are there from (0, 0, 0) to
(a, b, c) if we are only allowed to increase one of the coordinates by one at each
step?

Homework Equations

The Attempt at a Solution

This problem is easy for the path between (0,0) to (a,b)
because you can make the path into a binomial sequence, where if you increase a, then that will be a one, and if you increase b, then it will be a zero.

so for two:
total number of digits in the binary sequence= a+b
where a= # of 1s and b=#of 0's

So the total # of paths=
(a+b) choose a, which is equivalent to (a+b) choose b.

However for (0,0) to (a,b,c)
the total digits in the binary= a+b+c
and define the number of ones as a
and non-ones (zeros)= a+b

which yields:
Total path #s= (a+b+c) choose a.

I think the answer is:

((a+b+c) choose a)((b+c) choose b)

However, how do I explain that? Can i define c as the number of two's in the binary sequence? I thought binaries only had ones and zeros, that is why I haven't done that. Any help would be great!

 

[Elucidus]

This can be done using a multinomial formula.

But yes

$$\left( \begin{array}{c}{a+b+c}&{a}\end{array} \right) \left( \begin{array}{c}{b+c}&{b}\end{array} \right)$$

is fine.

--Elucidus