All Questions
Tagged with binomial-coefficients combinatorics
3,233
questions
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Closed form for nested sum involving ratios of binomial coefficients
I ran into the following nested sum of binomial coefficients in my research, but I couldn't find the closed form expression for it. I looked at various sources and still couldn't find the answer. So I ...
3
votes
1
answer
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Combinatorial Proof of the $\sum_{k=1}^{n} k^2 \binom{n}{k} = n(n + 1) 2^{n - 2}$ [duplicate]
How to prove that $$\sum_{k = 1}^{n}k^2\binom{n}{k} = n(n + 1)2^{n -2}$$ in an combinatorial way?
I have a algebraic proof method, but I don't know how to use a combinatorial proof method to do it.
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Evaluating Difference of Product of Binomial Coefficients
As part of my project I'm asked to evaluate the positivity of the following difference: $$\binom{l_1+t-1}{l_1}\binom{l_2+m+t-1}{l_2+m}\sum_{j=0}^{t}\binom{t-j+m}{m}\binom{j+l_1}{j}\binom{j+l_2}{j}-\...
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Spivak Exercise, Prove Vandermonde's Identity $\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$ [duplicate]
Prove that
$$\sum_{k=0}^{l}\binom{n}{k}\binom{m}{l-k}=\binom{n+m}{l}$$
Hint: Apply the binomial theorem to $(1+x)^n(1+x)^m$
Proof:
Following Spivak's advice, we have
$\sum_{k=0}^{n}\binom{n}{k}x^k=(...
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Choosing an ordered triplet of non-negative integers $(m_1, m_2, m_3)$ such that $m_1 + m_2 + m_3 = n$.
Define
$$A = \{ (m_1, m_2, m_3) : m_1 \geq 0, m_2 \geq 0, m_3 \geq 0, m_1 + m_2 + m_3 = n\}$$
Given that $n \geq 0$ and $n, m_1, m_2, m_3 \in \mathbb{Z}^+$, then why it is the case that
$$\vert A \...
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Trying to prove equivalence of combinatorial formula and nested summations
I��m sorry if this is a dumb problem, but I’m trying to get into mathematics and prove this but I’m only in 9th grade and haven’t found any sources on this:
Prove that $$ {n \choose r} = \overbrace{ \...
4
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3
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$(1-x)^{n+a} \sum_{j=0}^\infty \binom{n+j-1}{j}\binom{n+j}{a} x^j = \sum_{j=0}^a \binom{n}{a-j}\binom{a-1}{j} x^j$
Let $n$ and $a$ be natural numbers. How to prove the following for $x \in [0, 1)$?
$$
(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
$$...
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What is the name of this combinatorial identity?
In the course of my physics research, I appear to have stumbled onto the following combinatorial identity:
$${dn\choose m}=\sum_{\vec k} {n \choose \vec k}\,\prod_{j=0}^d {d \choose j}^{k_j},$$
where $...
- 1,835
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2
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Lower Bound on the ratio of binomial coefficients
Let $k,n,m$ be integers such that $k>n>m$. I am interested in providing a tight lowerbound on
$$
A(k,n,m)=\frac{\binom{k-m}{n-m}}{\binom{k}{n}}
$$
This term arises in a probability problem that ...
- 714
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164
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Estimate a sum of binomial coefficients
I should know this by the time, but: can someone tell me how to rigorously compute the leading order (including the constant) of the following sum:
$$\sum_{ 1\leq k \leq n/3 } {2 k \choose k} {n-2k-1 ...
- 1,363
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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 ...
- 3,234
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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....
- 2,145
2
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3
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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 ...
user1136752
1
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2
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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 Derive the Binomial Coefficient Upper Bound and Final Inequality in "Scheduling Multithreaded Computations by Work Stealing"?
In the paper Scheduling Multithreaded Computations by Work Stealing under the section "Atomic accesses and the recycling game", it mentions the binomial coefficient approximation:
$$ \binom{...
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