# Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far:

• induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates mentioned below and using identities of the binomial coefficient didn't get me anywhere

• managed to show $${n \choose k} \le {\left(\frac{ne}{k}\right)^k}$$ but it didn't quite help me

• Stirling estimates of n!: I was getting desperate, started googling

Can anyone help me out here? I really don't know how to go on with this task, since I can't even get a start. Thanks!

I can show the stronger statement ${20n \choose 10n}\ge {2n \choose n}^{10}$.

Suppose I have $20n$ books divided into 10 collections.

• The first one is the number of ways to choose $10n$ books out of my $20n$ books.

• The second one is the number of ways to choose $n$ books out of my
$2n$ books for every collection.

The first one is larger, since there if I choose $n$ books out of my
$2n$ books for every collection, I also have $10n$ books chosen out of $20n$.

It remains to be shown that ${2n \choose n} \geq {2n-1 \choose n-1}$, which is true because $${2n \choose n} = \frac{2n}n{2n-1 \choose n-1} = 2{2n-1 \choose n-1}$$

• Moral of this answer: Combinatorial proofs are sometimes very powerful. – wythagoras Feb 7 '16 at 13:17
• Thanks, that's perfect! I didn't even think of going over a stronger statement first, I guess I would've helplessly tried to show it for hours without your help. Now give me a few minutes to figure out how to mark your answer as correct and the question as done. – Tom P. Feb 7 '16 at 13:22
• @wythagoras: You’re welcome. (Nice answer.) – Brian M. Scott Feb 7 '16 at 13:45