If the numbers on the third diagonal of Pascal's Triangle are triangular numbers, what is the value of the $50$th triangular number? (The $n$th triangular number is $1+2+3+\ldots+n$.)
If the triangular numbers are found on the third diagonal of Pascal's Triangle, the triangular numbers are  \[\binom{2}{0}, \binom{3}{1}, \binom{4}{2}, \cdots,\]  where the $n$th triangular number is $\binom{n+1}{n-1}$. We're looking for the $50$th triangular number, which is  $$\binom{51}{49}=\frac{51!}{49!2!}\frac{51 \cdot 50}{2\cdot 1}=51\cdot25=\boxed{1275}.$$