Probability Question?

this is a problem of derangement/partial derangement

a derangement is an arrangement where no object is in its proper place,

e.g. if alphabetical arrangement ABCDE is "proper" , DCAEB would be a derangement

although a formula exists, (see source cited),

you may like to remember it as a recursive method

objects // derangements

2// BA :: D2 = 1

3// BCA, CAB :: D3 = 2 = 3*D2 - 1

4// BADC, BCDA, BDAC, CADB, CDAB, CDBA, DABC, DCAB, DCBA: D4=9=4D3 +1

continuing recursively, D5 = 5D4 -1 = 44

qa

exactly 1 match means exactly 4 deranged,

but any 4 of 5 may be deranged,

ie 5c4 = 5 ways for D4

so 5*9 = 45

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qb

D5 = 44

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