linear_combinatori_probabi
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Combinatorics I have been chasing for years: ($$U$$, the universe, is defined by $$S_0$$)

• $$\textrm{Exactly-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j}{k}}S_j.\quad$$ // All I have for you is a word, TENET.
• $$\textrm{At_Least-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j-1}{j-k}}S_j.\quad$$ // Pascal Explosion.

People proving it using mechanical-memory idea:

• $$\textrm{Let }m>k: \sum\limits_{l=0}^{k}\sum\limits_{j=l}^m=\sum\limits_{j=0}^m\sum\limits_{l=0}^{\min(j,k)}$$ // Independentness of addition.
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