Dinner-Diner Matching Probabilities

Consider a banquet at which n different entrees are served and k guests have requested each entree. Suppose the dinners are served randomly to the diners. What is the distribution of the number of matches of dinners to the diners who ordered them?

The following graphs illustrate the relationship between the dinner-diner matching probabilities and the normal distribution, and the relationship between the dinner-diner matching probabilities and the Poisson distribution. Sequence numbers refer to sequences listed in N. J. A. Sloane (2000), The On-Line Encyclopedia of Integer Sequences, published electronically at http://www.research.att.com/~njas/sequences/. The integer sequences are the numerators of the discrete probability distributions, laid out in triangular arrays.

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