Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician,

How can I prove the inequality
77!<88!

without complicated calculus? I tried and finally obtained just 17ln(7!)<18ln(8!)

Answer

Your inequality is equivalent to
(7!)8<(8!)7
divide it by (7!)7, and get
7!<87
and this is clear, since 17<88

Attribution
Source : Link , Question Author : Gwydyon , Answer Author : Crostul

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