Which statement best defines $n!$ ($n$ factorial)?

The sum of all positive integers from $1$ to $n$, i.e., $1+2+\cdots +n$

The product of all positive integers from $n$ down to $1$

The number of combinations of $n$ objects taken at a time when order does not matter

The sum of all positive integers from $1$ to $n$ multiplied by $n$, written as $(1+2+\cdots +n)n$