Quod Erat Demonstrandum

2008/02/19

A beautiful question in binomial theorem

Filed under: Additional / Applied Mathematics,HKALE,HKCEE,Pure Mathematics — johnmayhk @ 5:02 下午

證明:對任何正整數 m,n 恒有

\frac{(m + n)!}{(m + n)^{m + n}} < \frac{m!}{m^m}\frac{n!}{n^n}

如何證?MI?

看!對任何正整數 m,n 恒有

(m + n)^{m + n} > C_m^{m + n}m^mn^n

移項後,證畢。

美麗的題,美麗的解。講完,做o野。

來源:普特南(Putnam)2004 年考題

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