Say there are a objects of one kind, b of second and c of a third and so on. p particular objects in them, say x of type a, y of type b, z of type c and so on, should never be adjacent to each other. This can be solved by taking gaps or by inclusion-exclusion, but the process takes too much time. As an example Say, for example, in the word Preperation, the r, e, p should always be separated. Here, x=y=z=2. Is there a general formula for such a problem. I tried deriving one from inclusion-exclusion but didn't make progress.
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Does this solve your problem ? You get a variety of options ! https://math.stackexchange.com/questions/2626500/permutations-and-combinations-pens-of-same-color/2631651#2631651 – true blue anil Aug 13 '23 at 09:20