_k=0^6^51-kC_3 is equal to

Question

$\displaystyle\sum_{k=0}^{6}{}^{51-k}C_3$ is equal to

Accepted Answer

$$
\begin{aligned}
& \sum_{\mathrm{k}=0}^6{ }^{51-\mathrm{k}} \mathrm{C}_3 \\
& ={ }^{51} \mathrm{C}_3+{ }^{50} \mathrm{C}_3+{ }^{49} \mathrm{C}_3+\ldots+{ }^{45} \mathrm{C}_3 \\
& ={ }^{45} \mathrm{C}_3+{ }^{46} \mathrm{C}_3+\ldots \ldots+{ }^{51} \mathrm{C}_3 \\
& ={ }^{45} \mathrm{C}_4+{ }^{45} \mathrm{C}_3+{ }^{46} \mathrm{C}_3+\ldots . .+{ }^{51} \mathrm{C}_3-{ }^{45} \mathrm{C}_4 \\
& \left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}\right) \\
& ={ }^{52} \mathrm{C}_4-{ }^{45} \mathrm{C}_4
\end{aligned}
$$

Solutions · 1