If _k=1^10k^2(^10C_k)^2=2200L, then L is equal to _______.

Question

If $\displaystyle\sum_{k=1}^{10}k^2\left({}^{10}C_k\right)^2=2200L$, then $L$ is equal to \_\_\_\_\_\_\_.

Answer: 221

Accepted Answer

$$\sum_{k=1}^{10}\left(k\cdot{}^{10}C_k\right)^2=\sum_{k=1}^{10}\left(10\cdot{}^9C_{k-1}\right)^2=100\sum_{k=1}^{10}\left({}^9C_{k-1}\right)^2=100\cdot{}^{18}C_9$$

$$2200L=100\times48620\quad\Rightarrow\quad L=221$$

Solutions · 1