Question

Rezolvati urmatoarea ecuatie:
$C^{x}_{7} + C^{x-1}_{7}=28$

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

$\textbf{Proprietati:}\\C_n^k = C_{n-1}^k+C_{n-1}^{k-1}$

$C_{n}^k = \frac{k+1}{n-k}\cdot C_{n}^{k+1}$

$C_n^k = C_n^{n-k}\\\\\textbf{Rezolvare:}\\C_7^x+C_{7}^{x-1} = 28$

$\begin{aligned}\Rightarrow C_{8}^x &= 28= 28\cdot C_{8}^0 = 28\cdot \frac{0+1}{8-0}\cdot C_{8}^1 = \frac{28}{8}\cdot \frac{1+1}{8-1}\cdot C_{8}^2 = \\&=\frac{28}{8}\cdot \frac{2}{7}\cdot C_{8}^2 = 1\cdot C_{8}^2 = C_{8}^2\end{aligned}$

$\Rightarrow C_{8}^x = C_{8}^2 = C_{8}^{8-2} = C_{8}^6\\ \\ \Rightarrow C_{8}^x = C_{8}^2 = C_{8}^6\\\\ \Rightarrow \boxed{x\in \{2;6\}}$