Can you explain the complexity of this problem?

#1

lg (n!) = ………………?

#2
#3

I checked this out earlier but did not clearly understand.

#4

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itni samjh ayi
?
n* logn kaise aya
basically
pehle we opened n! as 1 .2 .3.4.5.6…n
right log(a.b) = log a + log b
so log(1 .2 .3.4.5.6…n) = log 1 + log 2 + log 3. … logn
so now the authors says
this is less than
logn + log n + log n …log n

now this log n factor is written n times
n * log n

hence the time complexity in worst case is n * log n