jmc

We will break this into three cases.

Case 1: numbers of the form $xyx$ ($x\ne 0$).

Any pair of nonzero digits has a corresponding palindrome ($xyx$) mountain number, so the number of these is $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$.

Case 2: numbers of the form $xyz$ ($z\ne 0,x\ne z$).

Any group of three nonzero digits ($y>x>z>0$) has two corresponding mountain numbers ($xyz$ and $zyx$), so the number of these is $2\times \left(\genfrac{}{}{0ex}{}{9}{3}\right)=168$.

Case 3: numbers of the form $xy0$ ($x\ne 0,y\ne 0$).

Any pair of nonzero digits has a corresponding mountain number in the form $xy0$, so there are $\left(\genfrac{}{}{0ex}{}{9}{2}\right)=36$ of these.

So the total number of mountain numbers is $36+168+36=\boxed{240}$.