Comments by "Jeff Huffman" (@tejing2001) on "The Riddle That Seems Impossible Even If You Know The Answer" video.

  1.  @SSkinner901  The chance of a random permutation of n things having a loop of k length is 1/k if and only if n < 2k. It works because the length of the loops precludes having more than one in a given permutation. As soon as you can fit 2 loops of that length in a single permutation, things get a lot harder to count properly. Fortunately in this case you can find the answer you're looking for without solving that problem. EDIT: To elaborate, the number of permutations of n things is obviously n!. The number of permutations of n things with a k-cycle (n < 2k) is the number of ways to make a k-cycle with a "marked" start point in n things (n!/(n-k)!) divided by the number of possible start points (k) times the number of ways to permute the remaining n-k items ((n-k)!). Putting that together is n!/(n-k)!/k*(n-k)! which reduces to n!/k. That's 1/k of the total number of permutations, n!. Also, none of these counts for distinct k overlap, since k+k' is always bigger than n and thus the 2 cycles can't coexist in the same permutation, so adding up those counts is fine.
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