I draw numbers 1 through k (k≤10) out of a hat ten times at random, replacing the numbers after drawing them. If I disregard the case where I draw "1" all ten times, explain why the number of possible sequences is divisible by 11. (Result by a calculator is insufficient because anyone can do that easily.)

Now if I change the number '10' to another integer n in the above paragraph, can I still have a similar result; i.e., the total possible number of configurations is divisible by n+1? Does this work for all integers n? If so, prove it; if not, find all integers n it works for.

...but I'll be having fun watching you.

I smelled something, looked it up, and I had smelled correctly -- but I'm not giving it away this early.

Let's just say it is a theorem by a famous mathematician whose other main theorem was proven only recently.

But it is nothing nasty - it can actually be proven in an elementary fashion, without background knowledge and without writing pages of formulas.  Have fun - I sure will.

Posted by vswitchs on 2006-09-04 12:50:18