# Tagged Questions

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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### Classes of $E=\{ 1,2,3 \}$

Question $2$: If $E=\{ 1,2,3 \}$ Determine the classes of all $E$ elements of E for each permutation $E=\{ 1,2,3 \}, \quad \Omega_{\sigma}(x)=\{ \sigma^{m};m\in \mathbb{Z} \}$ My Thoughts ...
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### How many strings possible with atmost 2 distance away?

So my question is you have given a string of n length , say abcd (n=4) . so how many unique strings of same length (n=4) possible such that they become at most 2 distance away from given string ? ...
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### Counting - Permutations & Combinations

Two series of a question booklet for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical ...
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### Coupon Collector Problem with packs

Given the coupon collector's problem, the expected number of coupons is calculated as follows: $E[X] = N \sum_{i=1}^N \frac{1}{i}$ This assumes we can draw one coupon at a time. Let's assume one ...
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### When the product of any two consecutive digits in the number is a prime number

I came across a question today. The number of 10-digit numbers such that the product of any two consecutive digits in the number is a prime number, is? As much as I know the product of any two ...
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### If $G = R \rtimes G_{\alpha}$ and $R$ is non-abelian of order $27$ and exponent $3$, then $G_{\alpha} \ncong D_{12}$

Suppose that $G = R \rtimes G_{\alpha}$ is a finite permutation group on $\Omega$ where $R$ is the non-abelian subgroup of order $27$ and exponent $3$ (see here). Suppose that $G_{\alpha} \cong D_n$, ...
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### Multinomial combinations with inequalities

The number of ways you can arrange $N$ balls into $m$ bins such that each bin has $\{n_1, n_2,...,n_m\}$ balls ($n_1+n_2+...+n_m=N$) is the standard multinomial $$\frac{N!}{n_1!n_2! \dots n_m!}$$ ...
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### In how many ways $7$ pencils can be distributed to $5$ children such that each child can get any number of pencils?

Q - In how many ways $7$ pencils can be distributed to $5$ children such that each child can get any number of pencils? I am little confused whether the answer is $5^7$ or $7^5$ ? I think every ...
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### prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$.
Let $X:S_n → GL_n(\mathbb{R})$ be the defining representation of $S_n$. Prove that $\det X(\pi) = \operatorname{sgn}(\pi)$ for all $\pi \in S_n$. attempt: I was thinking in trying to use \$X(e) = ...
This might be trivial but I am still confused about it. I am having some trouble in interpreting this question. My approach was to simply convert the question into the equation $$a + b + c = 5$$ ...