Find the number of ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c < 343$

The problem is that if $c=3$ you need $30a+50b \lt 133$. In that case $b$ can only be $1$ or $2$. If $b=2$ you must have $a=1$, while if $b=1$ you can have $a=1$ or $2$. The choices are not independent, so you cannot multiply.

You can just continue the casework. There are not too many cases for $b,c$.