25. In how many ways can the number $n$ be represented as the sum of three positive integer addends, if representations differing in the order of the addends are considered different?
25. The problem reduces to finding the number of positive integer solutions to the equation $x+y+z=n$.

First, note that the equation $x+y=l$ (where $l$ is a positive integer) has $l-1$ positive integer solutions. Indeed, in this case, $x$ can be equal to $1,2, \ldots, l-1$ (it cannot be equal to $l$, because then $y$ would not be positive); $y$ is determined from the equation and will also be a positive integer in these cases.

Now let's move on to our equation:

$$
x+y+z=n
$$

Obviously, $x$ can be equal to $1,2,3, \ldots, n-2$ (but it cannot be equal to $n-1$, because $y$ and $z$ are at least 1, and $n-1+1+1>n$); $y$ and $z$ satisfy the equation

$$
y+z=n-x
$$

Therefore, for a fixed value of $x$, we have $n-x-1$ possibilities for the values of $y$ and $z$. Hence, the total number of different solutions is

$$
\begin{aligned}
&(n-2)+(n-3)+\ldots+[n-(n-2)-1]= \\
&=(n-2)+(n-3)+\ldots+1=\frac{(n-1)(n-2)}{2}
\end{aligned}
$$