# Displacements on a grid

• 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 ?