- #1
jostpuur
- 2,116
- 19
Here's a nice problem: Prove that
[tex]
\frac{k!}{k^k} \leq \frac{(k-j)!}{(k-j)^{k-j}} \frac{j!}{j^j}
[/tex]
for all natural numbers such that [itex]0\leq j\leq k[/itex]. (Convention: [itex]0^0=1[/itex].)
I proved this myself, so I'm not asking for help any more. I merely decided to mention this problem to those of you who seek challenges
This claim seems to be so strong, that it cannot be proven with Stirling approximation. It will only tell you that the both sides of the inequality are roughly the same. You need to do some work before getting the precise result.
[tex]
\frac{k!}{k^k} \leq \frac{(k-j)!}{(k-j)^{k-j}} \frac{j!}{j^j}
[/tex]
for all natural numbers such that [itex]0\leq j\leq k[/itex]. (Convention: [itex]0^0=1[/itex].)
I proved this myself, so I'm not asking for help any more. I merely decided to mention this problem to those of you who seek challenges
This claim seems to be so strong, that it cannot be proven with Stirling approximation. It will only tell you that the both sides of the inequality are roughly the same. You need to do some work before getting the precise result.
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