query: How do I prove this combinatorics formula? $ 3^n = \sum_{k=0}^n\binom{n}{k}2^k $
positive_passages.docid: 19776
positive_passages.text: Simplify a combinatorial sum
negative_passages.docid: 130223
negative_passages.text: Proving a special case of the binomial theorem: $\sum^{k}_{m=0}\binom{k}{m} = 2^k$
negative_passages.docid: 113321
negative_passages.text: Hint proving this $\sum_{k=0}^{n}\binom{2n}{k}k=n2^{2n-1}$
negative_passages.docid: 48434
negative_passages.text: Prove that $\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$ using combinatoric arguments.
negative_passages.docid: 28027
negative_passages.text: Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
negative_passages.docid: 60860
negative_passages.text: Combinatorial proof of $\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$ and then use result to find a formula for $1^2 + 2^2 + \ldots + n^2$
negative_passages.docid: 59252
negative_passages.text: Show $\sum_{k=0}^{m} \binom{n-k}{m-k}=\binom{n+1}{m}$
negative_passages.docid: 126888
negative_passages.text: How can you show that $\binom {n}{7}=\sum_{k=7}^n \binom {k-1} {6}$?
negative_passages.docid: 155496
negative_passages.text: sum of combinations $\binom n{0} + \binom n{3} + \binom n{6} +... +\binom n{3k}$
negative_passages.docid: 75910
negative_passages.text: Combinatorial Proof of Combinatorial Identity involving $(-1)^k \binom {n-1}{k}$
negative_passages.docid: 136580
negative_passages.text: Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$
negative_passages.docid: 13236
negative_passages.text: Proving equality - a sum including binomial coefficient $\sum_{k=1}^{n}k{n \choose k}2^{n-k}=n3^{n-1}$
negative_passages.docid: 70128
negative_passages.text: Prove that $(fg)^{(n)} = \sum_{k=0}^n \binom{n}{k}f^{(k)}(x)g^{(n-k)}(x)$
negative_passages.docid: 65983
negative_passages.text: Combinatorial proof of binomial coefficient summation identity
negative_passages.docid: 116356
negative_passages.text: Combinatorial proof of $2 \cdot 3^0 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots 2 \cdot 3^{n-1} = 3^n-1$
negative_passages.docid: 108318
negative_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 71797
negative_passages.text: Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$