Algebra Maybe

$\dfrac{a+b+c}{a \times b \times c}\\ =\dfrac{a}{abc} + \dfrac{b}{abc} + \dfrac{c}{abc}\\ =\dfrac{1}{bc} + \dfrac{1}{ac} + \dfrac{1}{ab}$

Now, we know that the larger the values of the denominators, the smaller the value of the expression.

Thus, we are looking for the smallest possible values of $ab,\;ac$ and $bc$

Since we know that $a,\;b$ and $c$ are distinct integers from $1$ to $9$ inclusive, we pick the 3 smallest integers for $a,\;b$ and $c$ .

$a=1,\;b=2,\;c=3$ (or you can assign the values differently, it doesn't matter as long as there are these 3 numbers)

The maximum value of the expression

$=\dfrac{1}{2(3)} + \dfrac{1}{1(3)} + \dfrac{1}{1(2)}\\ =\dfrac{1}{6} + \dfrac{1}{3} + \dfrac{1}{2}\\ =\dfrac{1}{6} + \dfrac{2}{6} + \dfrac{3}{6}\\ =\dfrac{6}{6} =\boxed{1}$