jmc

Let $r_1$ and $r_2$ be the roots of this polynomial. Since $-\frac{b}{a}$ is the sum and $\frac{c}{a}$ is the product of the roots of $a{x}^2+bx+c=0$, we have $r_1+r_2=5$ and $r_1{r}_2=t$. Since $r_1$ and $r_2$ are positive integers, the only possible ordered pairs $(r_1,r_2)$ are $(1,4),(2,3),(3,2),$ and $(4,1)$. These produce the values of 4,6,6, and 4 respectively for $t$. Therefore, the average of the distinct possibilities, 4 and 6, is $\boxed{5}$.