Questions tagged [binomial-coefficients]
For questions involving the coefficients involved in the binomial theorem. $ \binom{n}{k}$ counts the subsets of size $k$ of a set of size $n$.
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Find the value of y from the expression $0 =\dfrac{1}{y^2} - x^2 + 2Cx^3 - \dfrac{4aC}{y^3}x$. [Ans. $ y = \dfrac{1}{x} + C - 2aCx $]
Here is my attempt.
$$\begin{align*}
& 0 =\dfrac{1}{y^2} - x^2 + 2Cx^3 - \dfrac{4aC}{y^3}x\\
\Rightarrow & \dfrac{1}{y^2} = x^2 - 2Cx^3 + \dfrac{4aC}{y^3}x \\
\Rightarrow & \dfrac{1}{y^2}...
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binomial distribution but sometimes the last outcome doesn't matter
Here's the motivation for my question: I'm designing an RPG. To simplify as much as possible, lets say my enemy has $h = 4$ HP and I deal $a = 1$ damage with every attack.
However, there's also a $p$ ...
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Iterated rascal triangle row sums
In this manuscript the authors propose the following conjecture (1)
\begin{align*}
\sum_{k=0}^{4i+3} \binom{4i+3}{k}_i &= 2^{4i+2}
\end{align*}
where $\binom{4i+3}{k}_i$ is iterated rascal ...
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Prove that this limit is equal to $\sqrt{2}$ for the function $f(x)=x^2-2$ for an arbitrary seed point $s$.
Mathematica knows that:
$$
s + \frac{1}{1-\lim_\limits{n\ \to\ \infty}\left[\frac{\displaystyle\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{f\left(k/n + s -1/n\right)}}{\displaystyle\sum _{k=1}^...
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consecutive binomial divisibility [closed]
Let s and t be positive consecutive integers such that $t=s+1$. Show that $s^{2n}+2nt-1$ is divisible by $t^{2}$
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What is $\sum_{m=0}^{\lfloor k/3\rfloor}{2k\choose k-3m}$ [closed]
With the floor function, I am not sure how to approach this.
Edit: I have a formula $\frac{1}{6}\left(4^k+2+3\frac{(2k)!}{(k!)^2} \right)$, but I got it purely from guess work.
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Given a set of integers, and the number of summations find the resulting frequencies
Given a set $X = \{x_1,x_2,...x_m\}\subset \mathbb{Z}$ and the number of allowed addends $N$. How can I find the frequency of every possible sum?
Example: $X = \{-1, 2\}$ and $N=3$ then every ...
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Is it true that $\sum\limits_{n\ge1}\binom{n+\frac1{4n}-1}n=\frac37$?
$$
\mbox{Is this closed form true ?:}\quad \sum_{n \geq 1}{n + 1/\left(4n\right) - 1 \choose n} =\frac37
$$
The series arises upon taking $\lambda=0$ in the identity from another post $$
\pi = 4+\sum_{...
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Summation of n-simplex numbers
Gauss proved that every positive integer is a sum of at most three triangular(2-simplex) numbers. I was thinking of an extension related to n-simplex.
Refer: https://upload.wikimedia.org/wikipedia/...
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Closed form for nested sum involving ratios of binomial coefficients
I ran into the following nested sum of binomial coefficients in my research, but I couldn't find the closed form expression for it. I looked at various sources and still couldn't find the answer. So I ...
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I want integration to turn $x^u$ into $\binom{m-u}{a-u}$
Like we have
$$
n! = \int_0^\infty e^{-t}t^n dt
$$
and
$$
\binom{n}{k}^{-1} = (n+1) \int_0^1 t^{n-k}(1-t)^k dt
$$
I would like to have a similar formula for the binomial coefficient $\binom{m-u}{a-u}$....
- 9,683
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Show $(n+1)^{r+1}-(n+1)=\sum_{k=1}^{r} \binom{r+1}{k}S_{r+1-k}(n)$ for $S_r(n)=\sum_{k=1}^{n} k^r$
Let:
$$S_r\left(n\right)=1^r+2^r+\ldots+n^r$$
for all $n,r\in\mathbb{N}_+$.
Show that:
$$\left(n+1\right)^{r+1}-\left(n+1\right)=\binom{r+1}{1}S_r\left(n\right)+\binom{r+1}{2}S_{r-1}\left(n\right)+\...
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an equation with binomial coefficients [duplicate]
I'd like to know how to prove the following property of binomial coefficientp.169 of Concrete Mathematics, by Ronald Graham, Donald Knuth, and Oren Patashnik
$$\sum_{0\leq k\leq l}\binom{l-k}{m}\binom{...
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Approximating a Binomial Expansion using Error Functions
Let $P_n(z) = \frac{1}{2^n}(1-z)^n = \displaystyle{\sum_{k=0}^n} \frac{{n \choose k}}{2^n} (-z)^k$.
If we let $X \sim B(n, \frac{1}{2})$ (that is X is a random variable following a binomial ...
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How can $\binom{n}{k}=\frac{n\cdot (n-1)\cdot ... \cdot (n-k+1)}{k!}$, with $k<0$ or $k>n$ be equal to $0$?
I can't mathematically understand how $\binom{n}{k}$, with $k<0$ or $k>n$, can be equal to $0$.
The part that I don't understand is (when $k < 0$) $\frac{n!}{k!\cdot (n-k)!}$, but $k!$ is ...
