2014 AMC 12B Problem 15

Since $k \ln k = \ln(k^k),$ the sum gives $p = \ln\left(\prod_{k=1}^{6} k^k\right),$ so $$e^p = 1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdot 5^5 \cdot 6^6.$$ The factors of $2$ come from $2^2$ (giving $2$), $4^4 = 2^8$ (giving $8$), and $6^6 = 2^6 \cdot 3^6$ (giving $6$). In total the exponent of $2$ is $2 + 8 + 6 = 16.$ Thus, the correct answer is C.