New posts in binomial-coefficients

Inductive proof for $\binom{2n}{n}=\sum\limits_{k=0}^n\binom{n}{k}^2$

combinatorics summation induction binomial-coefficients

Find the coefficient of the term $x^2$ in $\left(x+\frac 2x\right)^{4}$

binomial-coefficients binomial-theorem

A nice combinatorial identity: $\sum_{k=1}^{n-1}\frac{\binom{k-1}{n-k-1}+\binom{k}{n-k-1}}{\binom nk}=1$

combinatorics summation binomial-coefficients

Prove the identity $ \sum\limits_{s=0}^{\infty}{p+s \choose s}{2p+m \choose 2p+2s} = 2^{m-1} \frac{2p+m}{m}{m+p-1 \choose p}$

combinatorics binomial-coefficients

binomial congruence $\sum_{i=1}^{\frac{p-1}{2}}\binom{2i}{i}\equiv 0~or (-2)\pmod p$

number-theory binomial-coefficients

sum of product of three binomial coefficients

summation binomial-coefficients

Pine tree shaped in binomial coefficients and a proving the formula derived from the shape

combinatorics summation binomial-coefficients

Complicated sum with binomial coefficients

binomial-coefficients

How do I prove the negative binomial identity?

combinatorics binomial-coefficients

The limit $\lim_{r\to0}\frac1r\left(1-\binom{n}{r}^{-1}\right)$

limits binomial-coefficients gamma-function harmonic-numbers

Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n - 1}$ using the binomial theorem

discrete-mathematics summation binomial-coefficients binomial-theorem

Counting two ways, $\sum \binom{n}{k} \binom{m}{n-k} = \binom{n+m}{n}$ [duplicate]

combinatorics summation binomial-coefficients

How prove this inequality $(1+\frac{1}{16})^{16}<\frac{8}{3}$

inequality binomial-coefficients

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $ [duplicate]

binomial-coefficients binomial-theorem

proving $\binom{n-1}{k} - \binom{n-1}{k-2} = \binom{n}{k} - \binom{n}{k-1} $

combinatorics binomial-coefficients combinatorial-proofs

How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

linear-algebra algebra-precalculus binomial-coefficients

Finding coefficient of polynomial?

combinatorics functions polynomials binomial-coefficients

How can I prove this inequality for $n\geq 2$?

inequality binomial-coefficients

Why is $\sum \limits_{k = 0}^{n} (-1)^{k} k\binom{n}{k} = 0$?

combinatorics summation binomial-coefficients

matrix representations and polynomials

matrices polynomials representation-theory binomial-coefficients