# Questions tagged [binomial-coefficients]

For questions involving the coefficients involved in the binomial theorem. $\binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.

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### $(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j$

Let $n$ and $a$ be natural numbers. How to prove the following for $x \in [0, 1)$? $$(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j$$...
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### Deriving equation (3-66) of Papoulis and Pillai

$$\mbox{The identity is}\quad P_{2n+2} = P_{2n} + {2n \choose n}p^{n+2}q^n - {2n \choose n+1}p^{n+1}q^{n+1}$$ $\displaystyle\mbox{where}\ P_{2n} = \sum_{k=n+1}^{2n}{2n \choose k}p^kq^{2n-k} \quad$ ...
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### Identity regarding the sum of products of binomial coefficients.

Consider the following toy problem Person A and Person B have $n$ and $n+1$ fair coins respectively. If they both flip all their coins at the same time, what is the probability person B has more ...
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### Closed form for a sum of binomial coefficients

Let $m,n,r\in\mathbb{N}\cup\{0\}.$ I am interested in finding a closed form for the sum $$\sum_{i=0}^m{{n+i}\choose{r+i}}.$$ Let $f(m,n,r)$ denote the above sum. We may make a few trivial observations....
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### Proving $n! = \sum_{j=0}^n (-1)^j \binom{n}{j}(x+n-j)^n$ for any $x \in \mathbb{R}$ [closed]

How to prove the formula $$n! = \sum_{j=0}^n (-1)^j \binom{n}{j}(x+n-j)^n$$ for all $x \in \mathbb{R}$? I tried to use the binomial formula, but I am stuck with the problem. Thank you in advance for ...
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### Expansion of a function $(1-x)^{-n}$ [closed]

Write the expansion of the series: $(1-x)^{-n}$. I haven’t got the solution
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### What is the number of lattice paths of length 16 from the point (0,0) to (8,8) that go through (4,4) but don't go through (1,1), (2,2), (3,3)

what is the number of lattice paths of length 16 from $(0,0)$ to $(8,8)$ that go through $(4,4)$, don't go through $(1,1), (2,2), (3,3)$, and don't go over $y=x$? Here's what I tried: since we can't ...
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### Alternating sum involving binomial coefficients

I want to prove that $$\sum_{i=0}^{n}{n\choose i} \frac{\left(1 + \alpha i\right)^{n} \left(-1\right)^{n - i}}{n!} = \alpha^{n}.$$ This is a guess based on the computations for $n = 0,1,2,3$. Do you ...
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### How to prove that for $a,b \in \mathbb{C}$, $\sum\limits_{k=0}^n \binom{a}{k}\binom {b}{n-k}= \binom{a+b}{n}$?

This exercise came from Complex Analysis by Freitag and Busam 1.2.11. $$\binom{a}{n}:= \prod_{j=1}^n\frac{a-j+1}{j}$$ Show: $\sum\limits_{\nu=0}^{\infty}\dbinom{\alpha}{\nu} z^\nu$ is absolutely ...
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### Proving an Identity on Partitions with Durfee Squares Using $q$-Binomial Coefficients and Generating Functions

Using the Durfee square, prove that $$\sum_{j=0}^n\left[\begin{array}{l} n \\ j \end{array}\right] \frac{t^j q^{j^2}}{(1-t q) \cdots\left(1-t q^j\right)}=\prod_{i=1}^n \frac{1}{1-t q^i} .$$ My ...
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### Finding $\sum_{r=0}^n \binom{k}{2r}$ to solve a problem [Black Book by Vikas Gupta]. [duplicate]

The question goes as such: The number $N = {}^{20}C_{7}-{}^{20}C_{8}+{}^{20}C_{9}-{}^{20}C_{10}+.....-{}^{20}C_{20}$ is divisibly by: (A)$3$  (B)$4$  (C)$7$  (D)$19$ There are ...
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### Expressing a combination as sums of three terms or more, analogous to Pascal's identity

Are there any ways to express binomial coefficients as the sum of three or more terms, similar to how Pascal's Identity breaks a combination into the sum of two terms by relating two adjacent binomial ...
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Background information Bit flips Given a bit string, we say that bit flip happens when $0$ changes to $1$ or $1$ changes to $0$. To find bit flips, we can shift the string by $1$ and xor that new ...