I was trying to solve the following problem related to "counting cases":
Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the starting point $(0,0)$ to $(m,n)$ in unit steps, each step going upwards or to the right. Count the number of paths there exist to get from the initial point to $(m,n)$.
My attempt at a solution
To get to the point $m$ we have to move $m$ steps to the right and to get to the point $n$, $n$ steps upwards, so to get to $(m,n)$ we have to make a total of $m+n$ steps, $m$ to the right and $n$ upwards. Notice that there are different ways to do this task, for example, we can move first to the right $m$ steps and then when we are at $(m,0)$, we have to make $n$ steps upwards to get to $(m,n)$. Likewise, first we can get to the point $(0,n)$ by making $n$ steps upwards and then get to $(m,n)$ by advancing the $m$ remaining steps to the right.
It seems that up to some point, in each step you can choose "up" or "right", this stage at which you can't choose anymore would be when you are at a point of the form $(m,i)$ with $i<n$ or when you are at a point $(j,n)$ with $j<m$. I don't know how to translate this idea to a concrete numerical value, I would really appreciate if someone could help me.