I've been recently asked by one of my friends to prove an equation but still, I'm confused how to get it started tho.

log(n!)= θ(nlog(n))

Does anyone know how to help? I'll be very grateful if someone comes to reply to my issue.

Thanks in advance.

  • 1
    $\begingroup$ $\log(n!)=\sum_{k=1}^n\log{k}$ Use a Riemann sum. $\endgroup$ – saulspatz Feb 3 at 13:20
  • $\begingroup$ And now you just go for a worst case scenario, so you estimate every term by the upper bound $\log(n)$ $\endgroup$ – Wesley Strik Feb 3 at 13:23

$$O(\log{(n!)})$$$$=O(\log{(n(n-1)(n-2)...(2)(1))})$$$$=O(\log{(n)}+\log{(n-1)}+\log{(n-2)}+...+\log{(2)}+\log{(1)})$$$$=O(n\log{(n))}$$ As $n$ logarithms are added, we have a worst case time complexity of $O(n\log{(n))}$.

  • $\begingroup$ Thank you, sir. $\endgroup$ – Miller Feb 3 at 13:24
  • $\begingroup$ Will you accept my answer? $\endgroup$ – Peter Foreman Feb 3 at 13:25
  • $\begingroup$ Yes, surely. Thanks again. $\endgroup$ – Miller Feb 3 at 22:12

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