Fruits and Baskets Generalized

Discrete Mathematics Level pending

There are \(n\) baskets of fruits in a line, with \(n\) fruits also present. Each basket is designated to a certain fruit, and that fruit is placed inside it basket. How many ways can the fruits be rearrange such that no fruit is in its designated basket? Let \(a_1\) denote the answer if \(n = 1\), \(a_2\) denote the answer if \(n = 2\), and so on (\(a_n\) is the answer with \(n\) fruits and baskets). If I know the values of \(n-1, a_{n-1}\), and \(a_{n-2}\), how many calculations must I make to find \(a_n\).

Details and Assumptions

  • There is only one of each type of fruit.
  • Each basket must contain exactly one fruit.

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