jmc

If the triangular numbers are found on the third diagonal of Pascal's Triangle, the triangular numbers are $$\left(\genfrac{}{}{0ex}{}{2}{0}\right),\left(\genfrac{}{}{0ex}{}{3}{1}\right),\left(\genfrac{}{}{0ex}{}{4}{2}\right),\cdots ,$$ where the $n$th triangular number is $\left(\genfrac{}{}{0ex}{}{n+1}{n-1}\right)$. We're looking for the $50$th triangular number, which is $$\left(\genfrac{}{}{0ex}{}{51}{49}\right)=\frac{51!}{49!2!}\frac{51\cdot 50}{2\cdot 1}=51\cdot 25=\boxed{1275}.$$