Let the number of ivy, nightshade and triffid plants be $i$, $n$ and $t$ respectively.
$2i + 9n + 12t = 120$ and $i + n+ t = 20$, where $i> 0$; $n> 0$; $t> 0$.
Multiplying the second equation by $2$ and subtracting the new equation from the first:
$$7n + 10t=80$$
Thus $$7n = 10(8-t)$$
Therefore $n$ is a multiple of $10$ and since $1 \le n < 20$, $n=10$
Hence $8-t=7$ and therefore $t=1$.
This problem is taken from the UKMT Mathematical Challenges.