Summing The Triangle

Relevant wiki: Properties of Binomial Coefficients

If you sum all the elements in each row, you might notice a pattern:

$\begin{array}{c|c|c} \text{Row} & \text{Values in Row} & \text{Sum} \\ \hline 0 & 1 & 1 \\ 1 & 1+1 & 2 \\ 2 & 1+2+1 & 4 \\ 3 & 1+3+3+1 & 8 \\ 4 & 1+4+6+4+1 & 16 \\ \end{array}$

These sums are all powers of $2$ . In fact, the sum of all the elements in the $n^\text{th}$ row is $2^n$ . This is an application of the identity:

$\sum\limits_{k=0}^{n}{\binom{n}{k}}=2^n$

Thus, the sum of all the elements in the $12^\text{th}$ row is $2^{12}=\boxed{4096}$ .