I need help understanding why $ \left( \frac{n}{m} \right) ^k {{m}\choose {k}} \leq {{n}\choose {k}} $. Here m divides n, and k is a fixed small constant. I have tried expanding both sides, but not getting anywhere. Thanks.
Asked
Active
Viewed 557 times
1 Answers
0
The inequality can be rewritten as $$\frac mm\frac{m-1}m\cdots\frac{m-k+1}m\leq\frac nn\frac{n-1}n\cdots\frac{n-k+1}n.$$ Can you show that every factor on the left side is smaller then or equal to its corresponding factor on the right side?
Bart Michels
- 26,355
-
I wasn't looking at it the right way, thanks a lot! – user2945166 Jan 20 '14 at 19:40