jmc

The vertex of the parabola is $(3,2)$, so the equation of the parabola is of the form $$x=a(y-2{)}^2+3.$$ The parabola passes through the point $(1,4)$. Substituting these values into the equation above, we get $$1=a(4-2{)}^2+3.$$ Solving for $a$, we find $a=-1/2$. Hence, the equation of the parabola is given by $$x=-\frac{1}{2}(y-2{)}^2+3=-\frac{1}{2}(y^2-4y+4)+3=-\frac{1}{2}{y}^2+2y+1.$$ The answer is $\boxed{1}$.

Alternatively, the value of $x=a{y}^2+by+c$ is $c$ when $y=0$. The parabola passes through the point $(1,0)$, so $c=\boxed{1}$.