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How can you calculate the number of partial derangements of a multiset? For example, if I have the multiset {green, blue, blue, green, green, green, red, blue, red} and I want to find all derangements where 4 elements have not moved (4 fixed points) and $9-4=5$ elements have moved, I listed all of them and found there were 264 total. I found these links helpful:

How can I enumerate multiset partial derangements?

This would be a fully general form of partial derangements.

Peter Burbery
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    When you say "four fixed points", are you looking for all possible derangements where at least 4 are fixed? Or are the 4 fixed points set by the questioner? – David Raveh Jul 06 '23 at 22:42
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    Multisets doesn't care about order, hence the "set" part in the name, so ${1,1,2,2}$ is the same as ${2,2,1,1}$, son it makes no sense to talk about fixed points. You should consider tuples: $(1,1,2,2)\ne (2,2,1,1)$. – jjagmath Jul 06 '23 at 22:47
  • Order matters. Four points Don't move. Five points move. – Peter Burbery Jul 06 '23 at 23:32
  • Exactly four points don't move. If it was at least four points don't move then I have to do 0 points don't move one points don't move two points don't move 3 points don't move and four points don't move which is not what I'm focused on. – Peter Burbery Jul 06 '23 at 23:33

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