Let ^nC_r denote the binomial coefficient of x^r in the expansion of (1+x)^n.

Question

Let ${}^nC_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x)^n$. If $\displaystyle\sum_{k=0}^{10}(2^2+3k)\,{}^{10}C_k=\alpha\cdot3^{10}+\beta\cdot2^{10}$, $\alpha,\beta\in\mathbb R$, then $\alpha+\beta$ is equal to

Answer: 19

Accepted Answer

$$\sum_{k=0}^{10}(4+3k){}^{10}C_k = 4\sum_{k=0}^{10}{}^{10}C_k+3\sum_{k=0}^{10}k\cdot{}^{10}C_k = 4\times2^{10}+3\times10\times2^9$$

$$=2^{10}\left(4+\frac{3\times10}{2}\right)=2^{10}(4+15)=2^{10}\times19$$

Comparing with $\alpha\cdot3^{10}+\beta\cdot2^{10}$: $\alpha=0,\ \beta=19$, so $\alpha+\beta=19$.

Solutions · 1