(related to Proposition: Multinomial Coefficient)

The number of possible arrangements of \(n\) objects is given by the factorial \(n !\). By hypothesis, each of the \(k_i\) objects in each of the \(m\) different types (with \(k_1+k_2+\ldots+k_m=n\)) are indistinguishable. Therefore, in any of the \(n !\) different arrangements of \(n\) objects, there are \(k_i !\) indistinguishable arrangements of objects of the type \(i\) for \(i=1,\ldots,m\). By the fundamental counting principle, there are \(k_1 !k_2 !\cdot\ldots\cdot k_m !\) indistinguishable arrangements in total. It follows that the number of distinguishable arrangements of \(n\) objects of \(m\) given types with \(k_i\) objects in each type can be calculated by the formula \[\frac{n !}{k_1 !k_2 !\cdot\ldots \cdot k_m !}.\]

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  1. Matoušek, J; Nešetşil, J: "Invitation to Discrete Mathematics", Oxford University Press, 1998