How many possible ways to go right and up in an array
You can treat this problem as a combinatorics problem: How many ways are there to reach from $(0,0)$ to $(m,n)$?.

Your steps are either $\mathit{up}$ denoted by $1$ for clarity, or $\mathit{right}$, denoted by $0$

No matter what path you take, you must move exactly $n$ times $up$ and exactly $m$ times $right$.

it means you would, in total, perform $m+n$ moves

Since any move is essentially either $1$ or $0$, it can be reduced to the following questions: Given a sequence of moves $P$ of length $m+n$, how many such options are there for such a $P$ in which #$1=n$, #$0=m$ ?

Solution:


  $$\binom{n+m}{n} = \binom{n+m}{m} $$