query: How to map $\{0,1\}^n$ to $\binom{\{1, \cdots, n'\}}{n''}$ nontrivially?
positive_passages.docid: 143927
positive_passages.text: Binomial coefficients that are powers of 2
negative_passages.docid: 69498
negative_passages.text: What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?
negative_passages.docid: 10458
negative_passages.text: How to define a bijection between $(0,1)$ and $(0,1]$?
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: 158508
negative_passages.text: Combinatorial Proof for Binomial Identity: $\sum_{k = 0}^n \binom{k}{p} = \binom{n+1}{p+1}$
negative_passages.docid: 148949
negative_passages.text: Find the value of $\sum_{k=1}^{n} k \binom {n} {k}$
negative_passages.docid: 25004
negative_passages.text: Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
negative_passages.docid: 18046
negative_passages.text: Prove the following equality: $\sum_{k=0}^n\binom {n-k }{k} = F_n$
negative_passages.docid: 90293
negative_passages.text: How expand Binomial[n, k] for k >= 6?
negative_passages.docid: 62342
negative_passages.text: $x_1 \cdot \dots \cdot x_n=1 \implies x_1 + \dots + x_n \ge n$
negative_passages.docid: 9522
negative_passages.text: Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
negative_passages.docid: 26569
negative_passages.text: Counting binary operations on a set with $n$ elements
negative_passages.docid: 71797
negative_passages.text: Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
negative_passages.docid: 42606
negative_passages.text: Orthogonality for Binomial Coefficients
negative_passages.docid: 19914
negative_passages.text: Closed form for $\sum_{n=0}^{\infty} \binom{n+k}{k} x^n$ (Negative Binomial Theorem)
negative_passages.docid: 28027
negative_passages.text: Find $\sum_{k=0}^{n}4^k \binom{n}{k}$
negative_passages.docid: 131326
negative_passages.text: How to find $\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}$?