Rom

It's a miniature pony.  It's not mine.  Just a rando from youtube.

Apparently this site doesn't like us posting youtube links as my previous reply went into moderation limbo.

\(\left(x+\dfrac 1 x\right)^3 = \\ x^3 + 3x+ 3x^{-1}+x^{-3} = \\ 52 + 3\left(x + \dfrac 1 x\right)\\ u=\left(x+\dfrac 1 x\right)\\ u^3 - 3u - 52=0\\ (u-4) \left(u^2+4 u+13\right) = 0\\ \text{The left hand factor is zero at $x=4$, the right hand factor has complex roots}\\ \text{The solution you are after is $x=4$}\)

It has to do with what frequencies of light the object reflects well, which has to do with the atomic structure of the object at it's surface.

It's really quite a complicated question as to why this surface reflects red well while this one reflects green well and to really get an answer

you're going to have to delve into quantum electrodynamics which is not a topic for the faint hearted.

\(v = \alpha\begin{pmatrix}2\\-3\end{pmatrix},~\alpha \in \mathbb{R}\\ 2\alpha - 3\alpha = 3\\ \alpha = -3\\ v = \begin{pmatrix}-6\\9\end{pmatrix}\)

\(R(\theta) = \begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix}\\ \text{Now just plug in $\theta=\dfrac{\pi}{6}$}\)

it's not

(-1)^{(any even number)} = 1

take a hot bath and get a good night's sleep

you'll do fine

\(-5 \cdot -2 = 10 \text{ is the largest value for $b$}\)

Here is basically Melody's solution but rearranged a bit to make more intutive sense.

\(\text{First off there are $N=\dbinom{16}{4}\dbinom{12}{4}\dbinom{8}{4} =63063000 $ possible hands}\\ \text{First person gets their cards. There are 4 1's to choose from and $\dbinom{12}{3}$ ways to choose the other 3 cards}\\ \text{Second person has 3 1's and $\dbinom{9}{3}$ ways to pick the other cards}\\ \text{Third person has 2 1's and $\dbinom{6}{3}$ other cards}\\ \text{This gets us the total number of hands $n$ where each player has a 1 in their hand}\\ n = 4\dbinom{12}{3}\cdot 3\dbinom{9}{3}\cdot 2\dbinom{6}{3}\\ p = \dfrac n N = \dfrac{\dbinom{12}{3}\dbinom{9}{3}\dbinom{6}{3} 4!}{\dbinom{16}{4}\dbinom{12}{4}\dbinom{8}{4}} = \dfrac{8870400}{63063000 } = \dfrac{64}{455}\)