Bài toán 1: Chứng minh rằng:
$$\prod_{k=0}^{n}\binom{n}{k} \le \left(\frac{2^{n}-2}{n-1} \right)^{n-1};\forall n \ge 2$$
Làm bài 1 trước vậy
$$\binom{n}{0}=\binom{n}{n}=1 \Rightarrow \binom{n}{0}.\binom{n}{1}...\binom{n}{n}=\binom{n}{1}\binom{n}{2}...\binom{n}{n-1}$$
Áp dụng bất đẳng thức AM-GM ta có
$$\large{\prod_{k=0}^{n}\binom{n}{k} \le \begin{pmatrix}
\frac{\binom{n}{1}+\binom{n}{2}+...+\binom{n-1}{n}}{n-1}
\end{pmatrix}^{n-1}}$$
$$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}=2^n\Rightarrow \binom{n}{1}+\binom{n}{2}+...+\binom{n-1}{n}=2^n-2$$
Do đó $$\prod_{k=0}^{n}\binom{n}{k} \le \left(\frac{2^{n}-2}{n-1} \right)^{n-1};\forall n \ge 2$$