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## linear_combinatori_probabi $(function() {$(".js-rank-badge").addSpinner().load("/users/rank?userId=390226"); });

Learning: http://theanalysisofdata.com/ Combinatorically Happy List: ($$U$$ is the universe $$\Huge\textrm{:)}$$)

• $$\textrm{Exactly-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j}{k}}S_j.\quad$$ // All I have for you is a word, TENET.
• $$\textrm{Exactly-IEP(k)}=\sum\limits^{\large\color{blue}{\binom{m}{k}}}_{I\in S_k}\textrm{Exactly-IEP}_I(0).\quad$$ // Superposition.
• $$\textrm{At_Least-IEP}_U(k)=\sum\limits_{j=k}^m(-1)^{j-k}\color{blue}{\binom{j-1}{j-k}}S_j.\quad$$ // Combinatorial proof dream.

People proving it by mechanical-memory idea: (They try to be an expert but machine somehow)

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