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Could you help me?

I’m working in Sohrab Real Analysis and I have the following exercise:

Proof that for all ${n}\in\mathbb{N}$ we have

$\binom{n}{1}\binom{n}{2}\dots\binom{n}{n-1}\le(\frac{2^n-2}{n-1})^{n-1}$

I tried by induction over n and obtain the follow: If we suppose true for ${k}\in\mathbb{N}$ , then the in the inductive step I obtained

$\binom{k+1}{1}\binom{k+1}{2}\dots\binom{k+1}{k}\le{(k+1)}(\frac{2^{k+1}-2}{2(k-1)}-\frac{1}{k-1})^{k-1}$

And I don’t know how to follow.

Ricardo Mejías Constanzo
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1 Answers1

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Hint: Use the AM-GM inequality: $$\frac{a_1+a_2+\cdots+a_k}k\ge\sqrt[k]{a_1a_2\cdots a_k}$$ Think about what value for $k$ you should use and what $a_1, a_2,\ldots, a_k$ (all of them positive) must be.

(Second hint)

grand_chat
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