smc

There is no way to have a set with 0. If a set is to have its lowest element as 1, it must have only 1 element: 1. If a set is to have its lowest element as 2, it must have 2, and the other element will be chosen from the natural numbers between 3 and 12, inclusive. To calculate this, we do $\left(\genfrac{}{}{0ex}{}{10}{1}\right)$. If the set is the have its lowest element as 3, the other 2 elements will be chosen from the natural numbers between 4 and 12, inclusive. To calculate this, we do $\left(\genfrac{}{}{0ex}{}{9}{2}\right)$. We can see a pattern emerge, where the top is decreasing by 1 and the bottom is increasing by 1. In other words, we have to add $1+\left(\genfrac{}{}{0ex}{}{10}{1}\right)+\left(\genfrac{}{}{0ex}{}{9}{2}\right)+\left(\genfrac{}{}{0ex}{}{8}{3}\right)+\left(\genfrac{}{}{0ex}{}{7}{4}\right)+\left(\genfrac{}{}{0ex}{}{6}{5}\right)$. This is $1+10+36+56+35+6=\boxed{144}$.