Find the total number of non-negative integers less than $1000$ for which the sum of the digits is $10 .$

Rumour has it that Runnbeer, an alpha male, who easily gets to date the hometown hotties at the drop of his hat. Of late, he has been very promiscuous and ditched his two girls, who stay in the same city at two distinct locations Powai $\left(\mathrm{P}\right)$ and Qababchowk $\left(\mathrm{Q}\right).$

Today, Runnbeer has got a new date and so he wants to take his date from her apartment at Arthur road $\mathrm{A}$ to a restaurant at Churchgate $\left(\mathrm{C}\right).$ However, he does not want to go through the places $\mathrm{P}$ and $\mathrm{Q}$ as his deserted girls, Priyankhan and Qatrinah, stay at these two places only. Find the total number of the shortest paths that he can follow to reach $\mathrm{C}$ from $\mathrm{A}$ considering the roadmap given in the diagram.

How many distinct triangles are there in this figure?