HMMT 2017 A2

Find the value of

$$\sum_{1 \leq a<b<c} \frac{1}{2^{a} 3^{b} 5^{c}}$$

(i.e. the sum of $\frac{1}{2^{a} 3^{b} 5^{c}}$ over all triples of positive integers ( $a, b, c$ ) satisfying $a<b<c$ )