2
votes
0answers
47 views

How to prove these indentities?

How to prove these indentities? $\displaystyle \sum \limits_{k\geq0} {2n\choose 2k-1}{k-1\choose m-1}=2^{2n-2m+1}{2n-m\choose m-1}$ $\displaystyle \sum \limits_{k=0}^{m} {m\choose k}{n+k\choose ...
2
votes
1answer
33 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
votes
2answers
33 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
2
votes
1answer
45 views

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 ...
0
votes
7answers
138 views

Calculating $\binom{1}{2}$

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 ...
0
votes
1answer
56 views

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?
1
vote
1answer
58 views

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 ...
3
votes
1answer
80 views

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: ...
4
votes
1answer
77 views

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 ...
8
votes
3answers
426 views

Which is greater, $300 !$ or $(300^{300})^\frac {1}{2}$?

Which is greater among $300 !$ and $(300^{300})^\frac{1}{2}$ ? The answer is $300 !$ (my textbook's answer). I do not know how to solve problems involving such large numbers.
0
votes
1answer
76 views

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. ...
1
vote
0answers
88 views

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 ...
3
votes
2answers
420 views

German tank problem, simple derivation [duplicate]

I was reading the recent question on the German tank problem, and had trouble with one of the derivations in this section. $$\sum_{m=k}^N m \frac{\binom{m-1}{k-1}}{\binom N k} = ...
2
votes
1answer
67 views

Why is this formula for $(2m-1)!!$ correct?

Numerically calculating the sum of the squares of the $m$th row of Pascal's triangle, I found that for at least the first $10$ or so cases $$\sum_{i=0}^m \binom{m}{i}^2=\frac{(4m-2)!!!!}{m!}$$ Where ...
5
votes
3answers
233 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
3
votes
5answers
164 views

Can $\frac{n!}{(n-r)!r!}$ be simplified?

I'm trying to calculate in a program the number of possible unique subsets of a set of unique numbers, given the subset size, using the following formula: $\dfrac{n!}{(n-r)!r!}$ The trouble is, on ...
4
votes
4answers
423 views

Proving that ${n}\choose{k}$ $=$ ${n}\choose{n-k}$

I'm reading Lang's Undergraduate Analysis: Let ${n}\choose{k}$ denote the binomial coefficient, $${n\choose k}=\frac{n!}{k!(n-k)!}$$ where $n,k$ are integers $\geq0,0\leq k\leq n$, and ...
1
vote
4answers
421 views

Evaluate a sum involving n choose r

Evaluate: $\sum_{k=0}^6 (-1)^k \binom{6}{k}$ where $\binom{n}{r}= \frac{n!}{r!(n-r)!}$. I'm unsure how to compute the part with $\binom{6}{k}$, it should be something along the lines of ...
2
votes
3answers
113 views

Factorial Equality Problem

I'm stuck on this problem, any help would be appreciated. Find all $n \in \mathbb{Z}$ which satisfy the following equation: $${12 \choose n} = \binom{12}{n-2}$$ I have tried to put each of them ...
3
votes
2answers
61 views

Identity of binomial series with factorial.

I'm looking for a simple identity for the formula: $$ \sum_{k = 0}^{p} \binom{p}{k} \cdot k! \cdot x^k $$ In words, I have $p$ "players" who can choose to play or not (every player is represented by ...
2
votes
1answer
149 views

How to perform the summation/addition of binomial coefficients?

From my textbook: $$ \begin{align} \sum_{k=0}^n \binom {m+k}m &= \binom {m+n}m + \sum_{k=0}^{n-1} \binom {m+k}m\\\\\\ &= \binom {m+n}m + \binom {m+n}{m+1}\\\\\\ &= \binom {m+1+n}{m+1} ...
11
votes
1answer
129 views

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 ...
1
vote
0answers
46 views

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 ...
3
votes
0answers
127 views

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 ...
11
votes
1answer
589 views

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 \[ ...
0
votes
2answers
114 views

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 ...
4
votes
2answers
765 views

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 ...
16
votes
6answers
4k views

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 ...
6
votes
3answers
1k views

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 ...
9
votes
2answers
3k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
1
vote
1answer
105 views

Combination Problem with a Variable

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 ...