jmc

Firstly, observe that the second digit of such a number cannot be $1$ or $2$, because the digits must be distinct and increasing. The second digit also cannot be $4$ as the number must be less than $2400$, so it must be $3$. It remains to choose the latter two digits, which must be $2$ distinct digits from $\left\{4,5,6,7,8,9\right\}$. That can be done in $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=\frac{6\cdot 5}{2\cdot 1}=15$ ways; there is then only $1$ way to order the digits, namely in increasing order. This means the answer is $\boxed{\text{15}}$.