jmc

We begin by rewriting the equation as $2ab+10a-3b=222$. We can then use Simon's Favorite Factoring Trick by subtracting 15 from both sides of the equation to get $2ab+10a-3b-15=207$. This can be factored into $$(2a-3)(b+5)=207$$We know that the prime factorization of $207=3^2\cdot 23$ and that both $a$ and $b$ are positive integers, so the only possible solutions $(a,b)$ are $$(a,b)=\{(13,4),(6,18),(2,202),(3,64)\}$$Out of these, only $(6,18)$ meets the requirement that $a+b=24$. Thus, $ab=\boxed{108}$.