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comment What are the steps to solving this average distance problem?
The ordinary interpretation of average distance would give answer $1$.
comment for f(x,y,z) find point on surface nearest to origin
It you ask the question it will be quickly answered, but not by me (it is late). Lagrange multipliers, two constraints, will work. But it is really a linear algebra problem. If you want to set it up as a calculus problem, find parametric equations of the line of intersection of the two planes. The parametric equations will involve a single parameter $t$, and you will be minimizing a quadratic in $t$.
comment Variance of Normal Distribution
To me, $\text{normal}(a,b)$ means normal with mean $a$, variance $b$. Some people use the same notation for mean $a$, standard deviation $b$. It looks as if the person setting the problem uses the same interpretation as mine. If the problem is from a book or notes, the meaning of the notation was almost certainly explained.
comment example of an infinite group
@astudent: To show they have order $4$, all we need to do is to show that $i^4=1$, and $(-i)^4=1$, and that there is no $k$ with $1\le k\lt 4$ such that $i^k=1$ or $(-i)^k=1$. Just calculation. For showing there are no other elements of order $4$, consider the equation $z^4=1$, or equivalently $(z^2-1)(z^2+1)=0$. There are $4$ roots, $\pm 1$ and $\pm i$. But $1$ has order $1$ and $-1$ has order $2$, leaving only $\pm i$ with order $4$.
comment how can you tell if an angle is possible to construct?
Thanks, confused it with an entirely different problem.
comment Simplification of trigonometric expression using double and compound angle identities
The idea is fine, details not quite right. We have $4\sin x\sin 3x\cos x\cos 2x=\sin 4x\sin 3x$.
comment Even degree polynomial.
You are welcome.
answered Even degree polynomial.
comment Prove that the set of all binary sequences is uncountable
If you want to go back to basics, think in terms of Cantor diagonalization.
revised If I roll a 3-sided die $n$ times, what is the probability that each side shows up at least once?
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answered If I roll a 3-sided die $n$ times, what is the probability that each side shows up at least once?
comment find the number all $a$ less than 1000 such that the number of different prime divisors of them is $2$
Not pleasant. You will need a list of the primes less than $500$, and a systematic process.
comment $(X,Y)$ uniformly distributed over triangular region. Is my solution wrong?
$E(X)=0$ by symmetry, integration was not necessary. $E(Y)$ is correct, it is a geometric property of the median.
comment Find the coefficient of $x^{17}$ in the expansion of $(1 + x^5 + x^7)^{20}$.
One can use the binomial theorem, $(1+(x^5+7))^{20}$.
comment How to find a base number which has been used in an equation
One could guess one's way to the answer. Or else let the base be $b$. Then the left side is $3(b^2+3b+4)$.
answered solution sought for a function on integers no. 2
revised How to find $\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}$
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answered How to find $\lim_{n\to \infty}{\frac{n^{n-1}}{n!}}$
comment Indefinite integral - What am I doing wrong?
You can use Parts. First step was OK. Then note that $x^2=-(4-x^2)+4$, so our second integral is $-\int \sqrt{4-x^2}\,dx+\int \frac{4\,dx}{\sqrt{4-x^2}}$.
comment How to determine whether the series converges or diverges?
Ratio Test. Maybe first make things easier by observing that the $k$-th term is less than $\frac{3^k+k}{k!}$, indeed less than $\frac{3^k+3^k}{k!}$.