Combinations n things not all different

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SUMMARY

The discussion focuses on calculating combinations of n items taken r at a time, specifically when some items are identical. The proposed approach involves selecting k identical items (where 0 ≤ k ≤ r) and (r - k) non-identical items. This method leads to an expression that incorporates factorials, which can be simplified. The final step is to sum the results over the range of k, either from 0 to r or from 0 to p if r exceeds p.

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justwild
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Is there any formula which gives the combinations for n things (taken r at a time, r being less than n) which are NOT ALL different?
Say you have a total of n things and in those n things you have p things identical. Then you are required to select r things (r may be less or greater than p). Then what are possible ways of selecting r?
 
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Since nobody has answered yet, I will post my less-than-ideal answer which is more of a way that you could approach it rather than a guaranteed way of getting at the right answer.

You could first ask what the number of possible ways is, if you select [itex]0 \le k \le r[/itex] of the p identical ones and (r - k) of the non-identical ones. This should give you an expression involving loads of factorials that hopefully simplifies. Then sum over k = 0 to r (or k = 0 to p, if r > p).
 

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