The binomial coefficient $\dbinom{n}{k}$ can be defined in several equivalent ways for $n$ and $k$ non-negative integers: The number of subsets of size $k$ of a set of size $n$. Element $k$ of row $n$ in Pascal's triangle (counting the first element or row as $0$). The coefficient of $x^k$ in ...
3
votes
2answers
157 views
Why does $\binom{10}{7} = \frac{10!}{(10-7)!7!}$
We just learned that: $\dbinom{10}{7}= \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$, so that:
If you throw a dice 10 ...
8
votes
1answer
98 views
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
2
votes
5answers
113 views
Sum from 0 to n of $ n \choose i $? [closed]
Possible Duplicate:
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
Evaluation $\sum\limits_{k=0}^n \binom{n}{k}$
Is there a simple proof for this equality:
$$\sum_0^n {n ...
5
votes
1answer
93 views
proof that $1 = \sum\limits_{k=0}^n (-1)^k { 2n \choose n,k,n-k } \frac{n}{n+k}$
I'm looking for a proof of this identity:
$$
1 = \sum_{k=0}^n (-1)^k { 2n \choose n,k,n-k } \frac{n}{n+k}
$$
I'll take anything, but a combinatorial proof would be nice - all of the terms in the sum ...
3
votes
3answers
46 views
Proof of a Binomial Identity using a combinatorial argument
Question
Prove that if $k$ and $l$ are two positive integer with $k ≥ l$, then $\binom{2k}{2} =\binom{k−l}{2}+ \binom{k+l}{2}+ k^2 − l^2$
using a combinatorial argument.
I tried using Vandermonde's ...
2
votes
1answer
76 views
Sum of every $k$th binomial coefficient.
It is widely known that $$\sum_{m=0}^{n} {n\choose m} = 2^n$$
and that $$\sum_{m=0}^{\lfloor\frac{n}{2}\rfloor}{n\choose 2m} = 2^{n-1}$$ Both results can be proven by exploting the nature of the roots ...
1
vote
1answer
55 views
Binomial probability
Suppose that the probability of a company supplying a defective product is $a$ and the probability that the supplied product is not defective is $b$. Before each product supplied is released for ...
6
votes
3answers
61 views
How does Combination formula relates in getting the coefficients of a Binomial Expansion?
Sorry for the very basic question. I'm a programmer, not a mathematician. The title says it all but I'll just include a little background on why I asked the (stupid?) question.
I'd just finished ...
8
votes
0answers
112 views
Factorial canceling on expansion of binomial coefficents 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
\[
...
1
vote
1answer
90 views
Evaluating 'combinatorial' sum
Help me please to calculate the following sum. I have seen such kind of formulas in the papers related to combinatorics, specifically 'trees'. I am curious how to calculate or approximate this sum:
...
4
votes
1answer
64 views
Calculate $\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n\choose i}$
It is known that $\sum_{i=1}^n {n \choose i}=2^n$. I am wondering what would be the sum if we change the upper limit to $\sqrt n/2$, i. e. How to calculate$$\sum_{i=1}^{[\frac{\sqrt n}{2}]}{n \choose ...
4
votes
2answers
88 views
Combinatorial argument for this identity
Can you please give a combinatorial argument for the argument below?
$$\sum_{k=0}^n k^2 \binom{n}{k} = n(n+1)2^{n-2}$$
From RHS, I drew the following argument: There are $n+1$ people. In how ...
0
votes
0answers
60 views
Calculation of sum
I am wondering if it is possible to calculate or approximate the following sum
$$
\sum_{k=0}^l\frac{(l-2k)^p(2l+k(k-1))l^{k-1}}{(k+3)(k+2)}
$$here $p \geq 2$.
Thank you.
2
votes
1answer
107 views
Evaluation $\sum\limits_{k=0}^n \binom{n}{k}$
Please help me to evaluate combinatorially the following sum:
$$\sum_{k=0}^n \binom{n}{k}$$
Thank you.
0
votes
1answer
63 views
Simple question about an asymptotic equality
Could someone please explain the second equality in Conjecture 1.1: http://arxiv.org/pdf/math/0501313v2.pdf ? (reproduced below)
$(1+o(1))n^22^{1-n}=\left(\frac{1}{2}+o(1)\right)^n$
Initially, I ...
0
votes
0answers
40 views
Fisher information, unbiased estimator, and Cramer-Rao lower bound for the variance
Let $X\sim \operatorname{bin}(θ,n),\quad 0<θ<1$
a) Compute $I(θ)$, the Fisher information
b) Show that $Ψ(θ) = (X/(n-1))(1-(X/n))$ is an unbiased estimator $Ψ(θ)= θ(1-θ)$
c) Find the Cramer-Rao ...
5
votes
1answer
78 views
Intuition behind negative combinations
Take $\binom{n}{r}$. It denotes how in how many different ways you can choose $r$ elements from a set of $k$ elements. For case $\binom{4}{3}$ which evaluates to $\frac{4!}{3!(4-3)!}=4$, it perfectly ...
4
votes
2answers
123 views
Elementary bound of binomial coefficient
I'm working my way through an Erdös paper from the sixties and some of the elementary bounds he claims seem to be just beyond my reach. The expression looks horrendous but maybe there is a clever ...
1
vote
2answers
74 views
Why is the binomial coefficient related to the binomial theorem?
The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of ...
2
votes
2answers
95 views
Sum of cubes of binomial coefficients
I reduced a homework problem in combinatorics to giving an asymptotic estimate for
$\sum_{k=0}^n{n \choose k}^3$.
I assume Stirling's approximation can help, but I'm not experienced with making ...
2
votes
3answers
112 views
Combinatorial inequality $\binom{n}{j}\leqslant 2^n$
I was trying to prove (or to find a counterexample) of the following inequality:
$$\binom{n}{j}\leqslant 2^n$$
As I coudn't find a proof/counterexample, I tested some numbers and could see it ...
0
votes
1answer
50 views
Identity of binomial coefficients with a series
I never really used any series/infinite sums and now I should proove the following identity:
$$\sum\limits_{k=0}^{\infty}\binom{m}{k}\binom{n}{l-k}=\binom{m+n}{l}$$
Can you please explain me, how to ...
0
votes
1answer
109 views
Evaluate the Binomial Coefficient
Please help me evaluate the following Binomial coefficient using any known properties of Pascal triangle:
$$
\binom{52}{4} - \binom{47}{4} + \binom{47}{5}
$$
1
vote
1answer
54 views
Comparing two binomial coefficient sums
Let $j \in N, n\in N, n>1, q\geq 2$.
I would like to show that
$$
\sum_{j=\frac{n}{\ln n}}^{\sqrt n/2}(2j-n)^q{n \choose j}<\sum_{j=\sqrt n/2}^{{\frac n2}}(2j-n)^q{n \choose j}
$$
Any help would ...
1
vote
2answers
115 views
A sum involving powers of binomial coefficients.
Find the formula for the following sum of binomial coefficients:
$$ \sum_{m\ge 0} (-1)^m {\binom{n}{m} }^3 .$$ Could you find the formula for $\sum\limits_{m\ge 0} (-1)^m{\binom{n}{m}}^4$?
1
vote
0answers
61 views
A Curious Binomial Coefficient Sum
Let $k, l \leq n$ be non-negative integers. Does the following identity simplify?
\begin{align}
\sum_{j = 0}^{k} \binom{k}{j} \binom{j + n -l + 1}{n} = \binom{n - l + 1}{n} ...
1
vote
0answers
95 views
Calculation of a 'double' sum
Let $n \in N$ and $q\geq 2$.
I am trying to calculate the following sum:
$$
\sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!}
$$
Any help will be ...
3
votes
1answer
91 views
Evaluating a limit involving binomial coefficients.
If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer:
$$\lim_{n\rightarrow ...
2
votes
0answers
118 views
proof of formula and calculation sum
Show that following formula is true:
$$
\sum_{i=0}^{[n/2]}(-1)^i (n-2i)^n{n \choose i}=2^{n-1}n!
$$
Using formula calculate
$$
\sum_{i=0}^n(2i-n)^p{p \choose i}
$$
-1
votes
3answers
61 views
Prove a binomial inequality
I would like to prove the following inequality:
$$
{m+n \choose m} \ge \frac{(n+1)^m}{m!}
$$
Any hints?
2
votes
0answers
44 views
max number of times integer $>1$ appears in pascals triangle
$120, 210 ,3003$ appear $6$ times in Pascal's triangle.
$120={10\choose3}={16\choose2}={120\choose1}\\$
$210={10\choose4}={21\choose2}={210\choose1}\\$
...
1
vote
0answers
19 views
Inexpressibility of certain coefficients and discrete versions of Hölder's theorem
In an answer to a recent question, I noted that there were probably no explicit formulas for Stirling numbers (of the first kind, specifically) and speculated that this might be coupled to a sort of ...
2
votes
1answer
119 views
Summation: $ \sum \limits_{r=0}^n \frac{ \binom n r}{x+r} $
How to evaluate $$ \sum \limits_{r=0}^n \large \frac{\binom n r} {x+r} $$
I got this problem from a friend according to him, $ \binom n r$ is the coefficient of $(1+x)^n$. I am not sure how to ...
4
votes
2answers
318 views
computation of the sum
I am having trouble to compute the following sum:
$$
\sum_{k=0}^n(n-2k)^p \frac{{n \choose k}{2m-n \choose m-k}}{{2m \choose m}}
$$
Here $p\geq 2$.
To simplify the question, we can even assume that ...
11
votes
2answers
118 views
Proving a certain binomial identity with three parameters
I would like to prove the following identity:
$$\sum_{m\geq 0} (-1)^{i-m}{m+k \choose m} {i-1 \choose m-1}{m+k+1 \choose j} = \sum_{m\geq 0} {m+k \choose k}{k+1 \choose i-m}{k+1 \choose j-m}$$
for ...
0
votes
2answers
92 views
Prove that $n! = \sum_{i=0}^{n-1}{n \choose i}(n-i)^n (-1)^i$ [closed]
Possible Duplicate:
Proof of a combination identity:$\sum \limits_{j=0}^n{(-1)^j{{n}\choose{j}}\left(1-\frac{j}{n}\right)^n}=\frac{n!}{n^n}$
Prove that product of $n(n-1)(n-2)\dots(2)(1)$ ...
5
votes
1answer
58 views
An enlightening proof of a specific combinatorial identity
Concerns about the arithmetic genus of projective hypersurfaces led me to make the following combinatorial conjecture:
$${d-1\choose n+1} =\sum_{i=0}^{n+1} (-1)^{n+i+1} {d\choose i}$$
for $d \geq 1$, ...
3
votes
2answers
69 views
lacunary sum of binomial coefficients
I am sure there must be a known estimate for sums of the form:
$$
S_d(n)=\sum_{j=0}^n \binom{dn}{dj}
$$
where $d\ge 1$.
For $d=1$, the answer is obvious.
For $d=2$, the answer is here:
Sum with ...
1
vote
1answer
34 views
Probability question involving binomial theorom
I'm having difficulty solving this question :
In a multiple choice test there are 5 questions each with three possible answers. For each question a student chooses an answer at random. Find the ...
0
votes
2answers
68 views
Binomial Coefficients in the Binomial Theorem - Why Does It Work Question
to keep it simple: Given
$(a+b)^3=\binom{3}{0}a^3+\binom{3}{1}a^2b+\binom{3}{2}ab^2+\binom{3}{3}b^3$
Could you please give me an intuitive combinatoric reason to why the binomial coefficients are ...
6
votes
4answers
171 views
Showing that $\lceil (\sqrt{3} + 1)^{2n} \rceil$ is divisible by $2^{n+1}$.
I have a question which has fluxommed me and my pals for the past few days. Any help or solution is welcome
Show using Binomial theorem that the integer just after $(3^{1/2} + 1)^{2n}$ is divisble ...
2
votes
2answers
86 views
Combinatorial Proof for a $ p\mid\binom{p}{k} \ \ \ \ \ 0<k<p$ .
I'm looking for a combinatorial proof to the following statement:
$$ p\mid\binom{p}{k} \ \ \ , \ \ 0<k<p \ \ \ \ \ \ \text{and} \ \ p \ \text{is prime}.$$
Thank you.
0
votes
2answers
95 views
Simple Question About Binomial Theorem
On one of my calculus lectures I've seen the lecturer write:
$$(1+p)^n=1+np+\frac{n(n-1)}{2}p^2+\cdots+p^n$$
Could you please explain to me how did he get this equation?
Thank you very much.
1
vote
3answers
98 views
Sum with binomial coefficients
I'm repeating material for test and I came across the example that I can not do. How to calculate this sum:
$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}$?
3
votes
2answers
96 views
How to prove it by means of a combinatorial argument?(A combinatorial exercise) [closed]
Possible Duplicate:
Proof that $\binom{n}{\smash{0}}^2+\binom{n}{1}^2+\cdots+\binom{n}{n}^2=\binom{\smash{2}n}{n}$ using a counting argument
It is an exercise in a book on discrete ...
5
votes
3answers
149 views
Bounding ${n \choose k}$
Let $k > 0$, and $n > 2k$. Why is it necessarily true that
$${n \choose k} > \frac{n^k}{2^k k!}$$
And is the condition $n > 2k$ necessary for this inequality to hold?
1
vote
3answers
86 views
Proof that binomial coefficient is a natural number [closed]
Possible Duplicate:
Proof that a Combination is an integer
What is the proof that the binomial coefficient is a natural number?
$$k\ge0,n\ge k \implies {n \choose k} \in N,$$
I guess ...
0
votes
0answers
28 views
How do I find the variance of this binomial tree?
I have this tree:
which I used to get the expected value of a game. I want to construct a confidence interval for the expected value. How can I get the variance of this game?
The probabilities of ...
4
votes
1answer
162 views
Limit of alternating sum with binomial coefficient
I need to find a limit, or approximation for $\sum\limits_{k=1}^{n} (-1)^k {n \choose k} \log(a+bk)$ for, say, an $a,b\in (0,10)$. It is not so important what values $a$ and $b$ have. It would be ...
0
votes
1answer
40 views
Looking to generalize a binomial tree with some constraints.
I've got a set of sample data and I'm looking to see if it's possible to generalize a binomial formula to give a closed form solution to this. If not, would it be possible to write a program to do ...
