Tracking the number of factors of 2 and 3 on both sides of the equation:
On the RHS k! has approximately k factors of 2 and k/2 factors of 3. So a given factorial has about twice as many factors of 2 as 3.
(Both are roughly linear)
On the LHS, the 2's grow very fast. There are n(n-1)/2 factors of 2.
Quadratic growth
The 3's grow linearly again, this time to about 3k/4.
When n is small, the difference is small. n doesn't need to be very big to make the difference significant.
For example when n=6, there are 15 factors of 2 and only 4 factors of 3.
k! has 15 factors of 2 when k=16, but k! has 4 factors of 3 when k=9,10, or 11.
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Posted by Jer
on 2021-08-13 15:12:34 |
Poor attempt at an explanation
