[tex]f(x) = C_{3x+7}^{6x+2}\\ \\ 3x+7 \geq 6x+2 \Rightarrow 3x \leq 5 \Rightarrow x \leq \dfrac{5}{3} \\ \\ 3x+7 \geq 0 \Rightarrow 3x \geq -7 \Rightarrow x \geq -\dfrac{7}{3} \\ \\ 6x+2 \geq 0 \Rightarrow 6x \geq -2 \Rightarrow x \geq \dfrac{-2}{6} \\ \\ \Rightarrow -\dfrac{1}{3} \leq x \leq \dfrac{5}{3}\\ \\ f(x) = C_{3x+7}^{6x+2} \\ \\ \max(f(x)) = f\left(\dfrac{\dfrac{5}{3}+\dfrac{-1}{3}}{2}\right) = f\Big(\dfrac{2}{3}\Big) =[/tex]
[tex]= C_{3\cdot \frac{2}{3}+7}^{6\cdot \frac{2}{3}+2} = C_{9}^{6} = \dfrac{9!}{6!\cdot (9-6)!} = \dfrac{9!}{6!\cdot 3!} = \dfrac{6!\cdot7\cdot8\cdot9}{6!\cdot 3!} = 7\cdot 4\cdot 3 = \\ \\ = 84[/tex]
