What is the simple way to show that

Abbie Edwards

Abbie Edwards

Answered question

2022-03-12

What is the simple way to show that
Nlog{N}klog{k}logk!{N!}
I tried to use the factorial and the log rules but..
Thanks.

Answer & Explanation

Talaminiu2d

Talaminiu2d

Beginner2022-03-13Added 4 answers

The identity isn't true. Assuming the logs are natural, take k=e and N=2. We have
0.5099892elog2=2log2elogeloge!2=log2loge!0.47281
Edit: I see that you replaced an equality with an approximation. In this case, it depends on the relative size of N compared to k. To see this, just use Stirling's Approximation (on both of the factorials)
N!2πN(Ne)N.
That is,
logN!logk!NlogN+12(log2πN)-Nklogk+12(log2πk)-k

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?