Consider a rectangle of size $m \times n$. How many paths are there from the bottom left cell to the top right cell, making only steps to the right or up?
Choosing a path that only takes steps to the right or upwards to reach the top right of the rectangle $m \times n$ is equivalent to choosing among the $m+n$ steps in total the $n$ steps to be taken upwards. Thus, we have $\binom{m+n}{n}$ possible paths.

Another solution is as follows: we write on each cell $(a, b)$ of the rectangle the number of paths that start from the bottom right corner and end on this cell, only taking steps upwards or to the right. The integer written on the cell $(a, b)$ is the sum of the integers written on the cells $(a-1, b)$ and $(a, b-1)$, and the integer 1 is written on the cell $(0,0)$. Thus, we show by double induction on $a$ and then on $b$ that the number written on the cell $(a, b)$ is the integer $\binom{a+b}{b}$. Notice that here we observe the Pascal's Triangle, rotated, with its apex at the bottom right corner of the rectangle.