How many ways can one reach point (a,b) starting from (0,0)?

[geoffrey159]

Homework Statement

We fit the plane with a coordinate system, and we consider the set of points with coordinates in $\mathbb{N}\times\mathbb{N}$. To link two points in this coordinate system, we only allow unit displacements, and only increasing displacements.
In how many ways can one reach point $(a,b)$ starting from $(0,0)$ ?

Homework Equations

$d_i$ is the direction of axis $i$, $i = 1,2$

The Attempt at a Solution

There will be $a$ displacements in direction $d_1$, and $b$ displacements in direction $d_2$, for a total of $a+b$ displacements.
Since there are only two kinds of displacements, it fully describes a possible way to choose once and for all a direction, and to assign an order of appearance in $[[1..a+b]]$ for each displacements in this direction, among all displacements.
Let us choose $d_1$. The number of possible ways to link $(0,0)$ and $(a,b)$ is reduced to finding the number of parts of cardinal $a$ in $[[1..a+b]]$, which is $\binom {a+b} {a}$.
Do you agree with the reasoning ?

 

[Buzz Bloom]

Hi geoffrey:

Looks good to me.

Regards,
Buzz

 

[geoffrey159]

Ok, thanks !