1 .
Let $m \geq 1$ and $n \geq 1$ be integers. Consider a rectangle whose horizontal side has length $m$
and whose vertical side has length $n$. A path from the bottom-left corner to the top-right corner is
called valid, if in each step, it either goes one unit to the right or one unit upwards.
In the example below, you see a valid path for the case when $m = 5$ and $n = 3$.
How many valid paths are there?
In the example below, you see a valid path for the case when $m = 5$ and $n = 3$.
How many valid paths are there?
(a)
${m + n \choose n - 1}$
(b)
${m + n \choose n}$
(c)
$2^m + 2^n$
(d)
$2^{m + n}$