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 ... combinatorics combinations binomial-coefficients binomial-theorem- 65
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 ... permutations binomial-coefficients- 397
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 ... linear-algebra combinatorics convergence-divergence combinations binomial-coefficients- 31
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:\... combinatorics summation binomial-coefficients binomial-theorem- 11
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 ... combinatorics discrete-mathematics induction binomial-coefficients stirling-numbers- 11
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 ... combinatorics permutations binomial-coefficients- 9,673
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 ... combinatorics binomial-coefficients special-functions mathematical-physics laguerre-polynomials- 793
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 $$\sum_{k}\left(\begin{array}{c}l \\ m+k\end{... binomial-coefficients- 858
Generalised binomial coefficients make sense when characteristic is not 2.
$\newcommand{gbin}[1]{\binom{\frac{1}{2}}{#1}}\let\ge\geqslant$Consider the generalised binomial coefficient defined as $$\gbin{n} := \frac{\left(\frac{1}{2}\right)\left(\frac{1}{2} - 1\right) \cdots ... binomial-coefficients- 14.4k
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 $$\sum_{k=0}^n \binom{n}{k} = 2^n$$ ... summation algorithms binomial-coefficients hypergeometric-function telescopic-series- 2,447
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 ... combinatorics summation asymptotics binomial-coefficients- 503
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 ... real-analysis calculus combinatorics asymptotics binomial-coefficientsHelp 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 ... combinatorics graph-theory binomial-coefficients- 133
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 ... summation binomial-coefficients- 43
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 \... sequences-and-series recurrence-relations binomial-coefficients closed-form lucas-numbers- 5,144
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