Questions Tagged [Binomial-Coefficients]
Ask Question Coefficients Involved in the Binomial Theorem. $ \Dbinom{N}{K}$ Counts the Subsets of Size $K$ of a Set of Size $N$. 6,921 Questions More Bountied...
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Coefficients involved in the Binomial Theorem. $ \dbinom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.
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Find $X/1430$ when $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$
Let $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$, then what's the value of $X\over1430$?
I don't even know where to begin on this question. All solutions I've seen on various ...
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If the word “WOW” can be rearranged in exactly 3 ways (WOW, OWW, WWO), how many different arrangements of the letters in “MISSISSIPPI” are possible? [duplicate]
The total number of distinct arrangements which is $\frac{11!}{1!4!4!2!}=34650$
How is this calculated?
Is this a binomial coefficient?
I don't understand why the denominators are the size of the ...
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A tight upper bound on this Binomial sum
I have the following function:
$P(n)=q^n\sum\limits_{H=0}^{n-1}{{H+n-1\choose H}w^H}+w^n\sum\limits_{H=n}^{\infty}{{H+n-1\choose H}q^H}$, where $0<q<0.5<w<1$ and $q+w=1$.
My end goal is to ...
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Sum of binomial coefficients for a specific sum
I am trying to find the eigenvalues of a matrix, and the degeneracy of each eigenvalue is given by the following expression:
\begin{equation}
deg(2l)=4\sum_{\substack{\{0\leqslant 2i,2k \leqslant L^2:\...
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A proof for stirling numbers of the second kind... [closed]
So my Prof give me this Statement to proof and i have no idea how i could solve it tho.
$$S(k,n)=S(k−1,n−1) + n \cdot S(k−1,n)$$
My task is to prove it and as hint he said:
Use The binomial ...
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What is the flaw in this approach?
$12$ delegates exists in three cities $C_1,C_2,C_3$ each city having $4$ delegates. A committee of six members is to be formed from these $12$ such that at least one member should be there from each ...
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Combinatorial problem: triple binomial product related to squared Laguerre polynomials
Context
Hydrogenic wavefunctions [1] include a factor given by Laguerre polynomials [2]. These wavefunctions are often encountered in a first course in quantum mechanics. They also appear in ...
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Summation of products of binomial coefficients $\sum_{j=0}^{n-1} (-1)^j {j\choose n-s+r}{n\choose j-r}$
I'm trying to sum $$\sum_{j=0}^{n-1} (-1)^j {j\choose n-s+r}{n\choose j-r}$$ with $n,s,r \leq s$ are integers.
I can find the related identity stated as $$\sum_{k}\left(\begin{array}{c}l \\ m+k\end{...
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Generalised binomial coefficients make sense when characteristic is not 2.
$\newcommand{gbin}[1]{\binom{\frac{1}{2}}{#1}}\let\ge\geqslant$Consider the generalised binomial coefficient defined as
$$\gbin{n} := \frac{\left(\frac{1}{2}\right)\left(\frac{1}{2} - 1\right) \cdots ...
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Using Gosper's algorithm to obtain the WZ certificate of $\sum \binom{n}{k} = 2^n$
I'm not sure where my work is wrong, I'm not obtaining an answer, even though I know there should be one.
In order to obtain the WZ proof certificate for the sum
$$\sum_{k=0}^n \binom{n}{k} = 2^n$$
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What is $\sum_{k = 1}^n (k \log k)\binom{n}{k}$? If the exact answer is difficult to find, what is the tightest asymptotic upper bound?
While trying to solve the complexity of my program I came across the the following summation:
$$\sum_{k = 1}^n (k \log k)\binom{n}{k}$$
Could you please provide a solution to this sum. If it is ...
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Finding a closed form for a limit of a sequence
Consider a sequence $$u_n=\sum_{r=1}^{n} \frac{{n\choose r} f^{(r)}(1)}{(r-1)!} $$ where $f$ is an infinitely differentiable real valued function on $\mathbb
{R}$.
Question: Given that the limit ...
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Help with simplifying a combinatorial sum
I'm currently counting different types of coloured graphs and have arrived at a particularly long sum of binomial coefficients. Can someone with more experience with combinatorial identities see a way ...
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Find the closed form of summation of binomial coefficients
For positive integers $k$, I have gotten that $$\sum\limits_{i=0}^{k}\frac{(-1)^{k+i-1}}{i+1}\binom{k+i}{i}\binom{k}{i}=0.$$
But for positive integers $m$ with $1\leq m<k$, how can I get the closed ...
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Closed form representation for $\sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}k$
Answering some other question, I stumbled upon the following relationship:
For $n\in\Bbb N$ let
$$p_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}k$$
and let
$$a_n = p_n+p_{n-2}\quad \text{ if } n \...
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