query: How to prove if this $\sum_{l=0}^{n}\binom{n}{l}=2^{n}$ is valid for all $n\in \mathbb{N}$?
positive_passages.docid: 4991
positive_passages.text: Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$
negative_passages.docid: 108318
negative_passages.text: counting proof for $\sum_{k=0}^{n-1} 2^{k} = 2^n - 1$
negative_passages.docid: 129304
negative_passages.text: Alternative combinatorial proof for $\sum\limits_{r=0}^n\binom{n}{r}\binom{m+r}{n}=\sum\limits_{r=0}^n\binom{n}{r}\binom{m}{r}2^r$
negative_passages.docid: 65672
negative_passages.text: Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
negative_passages.docid: 104704
negative_passages.text: Evaluate $\sum\limits_{k=0}^n \binom{n}{k}$ combinatorially
negative_passages.docid: 19571
negative_passages.text: Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
negative_passages.docid: 133905
negative_passages.text: Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
negative_passages.docid: 17638
negative_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
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: 142994
negative_passages.text: How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?
negative_passages.docid: 50226
negative_passages.text: Prove the identity $\sum_{k=0}^{n}\sum_{r=0}^{k} \binom{k}{r} \binom{n}{k} = 3^n$
negative_passages.docid: 64378
negative_passages.text: Proving a binomial sum identity $\sum _{k=0}^n \binom nk \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}$
negative_passages.docid: 2885
negative_passages.text: Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
negative_passages.docid: 148949
negative_passages.text: Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
negative_passages.docid: 30672
negative_passages.text: Show that $\sum\limits_{k=0}^n\binom{2n}{2k}^{\!2}-\sum\limits_{k=0}^{n-1}\binom{2n}{2k+1}^{\!2}=(-1)^n\binom{2n}{n}$
negative_passages.docid: 116722
negative_passages.text: Simplifying $\big(\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}\big)/{\binom{M}{n}}$
negative_passages.docid: 116491
negative_passages.text: Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$