#901 · Combinatorics · Counting

Combinations with repeated items

Given $p + q + r$ objects where $p$ are identical, $q$ are identical (but different from the first $p$ ), and the remaining $r$ objects are all distinct, show that the total number of non‑empty selections (combinations) that can be formed is $(p + 1) (q + 1) 2^{r} - 1$ .

~5 minProof · checked by AIidentical objectscombinationscounting

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XI. Permutations and Combinations, Examples XI.b, No. 31

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