Chapter 5
Counting with Repetitions
In counting combinations and permutations, we assumed that we were drawing from a set in which all of the elements are distinct. Of course, it is easy to come up with a scenario in which some of the elements are indistinguishable. We need to know how to count the solutions to problems like this, also.
Summary.
The number of ways of choosing \(r\) objects from \(n\) types of objects (with replacement or repetition allowed) is \(\multiset{n}{r}=\binom{n+r-1}{r}\text{.}\)
The number of ways of arranging \(n\) objects where \(r_i\) of them are of type \(i\) (indistinguishable), is \(\binom{n}{r_1,r_2,\ldots, r_m}\text{.}\)
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Notation:
\(\displaystyle \multiset{n}{r}\)
\(\displaystyle \binom{n}{r_1,r_2,\ldots, r_m}\)