How do I Explain this Combinatory? $$ \sum_{k=1}^{100} {{100} \choose {k}}2^k=3^{100}-1 $$
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@ms._VerkhovtsevaKatya yes thank you ! – user12499410 May 01 '20 at 12:35
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6If explictly looking for a combinatorial explanation... Consider the number of ternary strings (strings made with digits 0,1,2) with at least one non-$2$. On the right, count directly. On the left, first pick the locations and then the values of each non-$2$, ranging over all possible numbers of non-$2$'s. – JMoravitz May 01 '20 at 12:35
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@JMoravitz that's good! – user12499410 May 01 '20 at 16:28