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There are m points on the circle. Choose k points among those m points. And k points should not neighbor each other.

My Idea is... there are :

  • k numbers of black stones
  • n-k numbers of white stones
  • place white stones on the circle (n-k-1)! b.c placing on the circle.
  • And remove orders on white stone by dividing (n-k-1)! by (n-k)!
  • And place black stones between white stones. ${n-k \choose k}$

So my answer is $\frac{1}{n-k} *$ $n-k \choose k$
But answer on my textbook is $\frac{n}{n-k} *$ $n-k \choose k$
I think I should multiply my answer by n b.c n points is already fixed on the circle and I didn't considered it when I place white stones on it, so I should multiply it by n.

Please give me your feedback or other solutions!

K.S Kim
  • 65
  • Is $m=n$? If $(m,k)=(2,1)$ then there are two choices, clearly. But your expression gives $\frac 11\times \binom 11=1$. So go through your argument line by line in this case to spot the error. – lulu Apr 03 '22 at 00:01

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