# Tagged Questions

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### Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
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### How to calculate this sum

How do you calculate this sum $\sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
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### Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...
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### Calculating $\displaystyle\sum_{i=1}^{n} \binom{i}{2}$

Show $\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to $i!/2!(i-2)!$ and ...
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### Evaluate a finite sum with four factorials

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate $$\sum^{n}_{i\mathop{=}0}\frac{1}{n+k+i}\cdot\frac{(m+n+i)!}{i!(n-i)!(m+i)!}$$ Any hints? I'm stuck on ...
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### Closed form of $n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$

$n$ is given, and it takes part in the following formula. $$n!\sum_{k=3}^{n-1}{{n-2}\choose{k-1}}$$ Is there a nicer way for expressing it? Without the summation sign?
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### Binomial coefficient properties

On "theoretical computer science cheat sheet" I found a special formula which is: $${n \choose k} = (-1)^k {k-n-1 \choose k}$$ But when I try to expand the value of ${k-n-1 \choose k}$ I have ...
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### Sum of fraction of factorials

Can anybody explain this? $$\sum\limits_{k=1}^{\frac{m-1}2}\frac{(2k)!(2m-2k)!}{(2k-1)(2m-2k-1)k!^2(m-k)!^2}=\frac{(2m)!}{(2 m-1)m!^2}$$ I did actually simplify this to: ...
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### Simple method for $\frac{(2n+1)!}{(n!)^{2}}$ divide $lcm(1,2,\ldots,2n+1)$

The question is to prove that $\frac{(2n+1)!}{(n!)^{2}}$ divides $lcm(1,2,\ldots,2n+1)$. This seems like it should be a simple question, but try as I might, I can't seems to find any way that does ...
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### factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
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### Theory behind multiplication & combinations?

If with the Binomial Coefficient we try to find the possible combinations $\binom{n}{k}$ where $n$ is equal to $k$ what is the theory behind factorial resulting in the correct solution? E.g. ...
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### Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
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### Can this product be written so that symmetry is manifest?

Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression $$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$ apparently computes the ...
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### Approximation of factorial - Stirling formula [duplicate]

Possible Duplicate: Elementary central binomial coefficient estimates How can I prove that $$\binom{n}{n/2} = \Theta\left(\frac{2^n}{\sqrt n}\right)$$ I tried with Stirlings ...
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### Prove: $\frac{(2px)!}{((px)!)^2}\equiv\frac{(2x)!}{((x)!)^2}\pmod{p^2}$

How can I prove the following, where $p$ is a prime and $x$ a positive integer? $$\dfrac{(2px)!}{((px)!)^2}\equiv\dfrac{(2x)!}{((x)!)^2}\pmod{p^2}$$ I'm not sure if it is actually true, but I tested ...
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### Factorial canceling on expansion of binomial coefficients on Concrete Mathematics

On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as: $\frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z}$ where \[ ...
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### What is the minimum number of friends Sally can have if she can invite a different group of friends to her house every night for a year?

The question: Sally has $N$ friends and likes to invite them over in small groups for dinner. She calculated that she can invite a different group of $3$ friends to dinner at her house every night for ...
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### Factorial division using Pascal's triangle.

I want to get values of factorial divisions such as 100!/(2!5!60!)(the numbers in the denominator will all be smaller than the numerator, and the sum of the ...
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### prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer

Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared. My thought process: The numerator is the product of the first n even ...
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### Inductive proof for the Binomial Theorem for rising factorials

I want to proove the following equality containing rising factorials $$(x+y)^\overline{n}\overset{(*)}{=}\sum_{k=0}^n\binom{n}{k}x^\overline{k}y^\overline{n-k}.$$ For $n=1$ this equality is ...
We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
I have the following problem: $_xC_6$ = $_xC_4$ I expand both sides to: $$\frac{x!}{[(x-6)!]6!} = \frac{x!}{[(x-4)]!4!}$$ Next I multiply both sides by the denominator of the right-hand ...