Problem 440

The third digit must be at least $3$ since the smallest possible sum of two
digits from $1$ through $9$ is $1+2=3$.

If the third digit is $3$, then
the combination must be $1\,2\,3$ in
that order.

If the third digit is $4$, then
the other two digits must be $1$ and
$3$ since these are the only two
distinct digits that have a sum of $4$. Thus, the only possible combination
with third digit $4$ is $1\,3\,4$. Note that $2\,2\,4$ is not allowed since the digits
must be distinct.

If the third digit is $5$, then
the first two must be either $1$ and
$4$ or $2$ and $3$. Continuing in this way, we can list
all possible combinations:

$1\,2\,3$, $1\,3\,4$, $1\,4\,5$, $1\,5\,6$, $1\,6\,7$, $1\,7\,8$, $1\,8\,9$,

$2\,3\,5$, $2\,4\,6$, $2\,5\,7$, $2\,6\,8$, $2\,7\,9$,

$3\,4\,7$, $3\,5\,8$, $3\,6\,9$,

$4\,5\,9$

There are $16$ possible
combinations.