# Questions tagged [products]

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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### Can I write $\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$? [closed]

Is product notation distributive. As in, can I write the following? $$\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$$
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### If $N$ and $K$ are normal subgroups of a group $G$ such that $G=MN$ and $M\cap N=\langle e \rangle$ then $G=M\times N$.

The following is an exercise in Hungerford's abstract algebra text. If $N$ and $K$ are normal subgroups of a group $G$ such that $G=MN$ and $M\cap N=\langle e \rangle$ then $G=M\times N$. If $G=S_3$ ...
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### How to prove $\prod_{k=0}^n\left(2-\frac{2k+1}{n}\right)=-\frac{(2n)!}{2^n n^{n+1}n!}$.

To finish a proof, I am stuck on the steps of getting from $$\prod_{k=0}^n \left(2-\frac{2k+1}{n}\right)$$ to the form $$-\frac{(2n)!}{2^n n^{n+1} n!}.$$ If it helps, the entire question as follows: ...
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### Prove that $\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$

For arbitrary $x$ and $1\leqslant m\leqslant n$, prove the following: $$\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$$ I'm looking for a proof that ...
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### The result of Big Pi notation "without any element". [duplicate]

This is such a simple question but I couldn't find the answer on the internet. What is the default result of the Big Pi notation when it happens to be applied to an empty set? Is it 1 or 0, or even ...
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### Closed-form value of a product $\prod_{i = 2}^n (1 + \frac{p}{i})$ with $0 \leq p \leq 2$

Let $f(n+1;p) = \prod_{i = 2}^n (1 + \frac{p}{i})$, where $0 \leq p \leq 2$ and $n \geq 2$ with $f(2;p)=1,\forall p$. We have $f(n;0) = 1$, $f(n; 1)=n/2$. I also can see that $f(n; 2) \leq n^2/4$. Is ...
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### How can I prove that this matrix is idempotent?

I have the following matrix $$A=\begin{equation} \begin{pmatrix} 0 & a & -b\\ -a & 0 & c\\ b & -c & 0 \end{pmatrix} \end{equation}$$ I have to prove that $M=A^2+I$ is ...
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### Geometric Mean To Calculate Event Probability vs Product Of Outcomes?

I have a problem where I need to calculate the sum of the probabilities of certain outcomes (events are A,B,C or D), but would like to do it in one formula. Currently I have the following data: ...
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