# All Questions

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### What is the logic behind the probability of getting 'four of a kind' in poker?

This hand ($5$ cards of $52$) has the pattern $AAAAB$ where $A$ and $B$ are from distinct kinds. The number of such hands is $\binom{13}{1} \binom{4}{4} \binom{12}{1} \binom{4}{1}$. The probability ...
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### Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
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### if $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$

I have a question $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$ in [0,$\infty$)? can we say that the $f(\sqrt{x})$ also can be infinitely differentiable in ...
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### Use Induction to prove $\forall m,n \in \Bbb Z_{\ge 0}, 1 +mn \leq (1 + m)^n$

Use Induction to prove: $$\forall m,n \in N, 1 +mn \leq (1 + m)^n$$ for integers $m,n\ge 0$. My biggest problem with this proof is ...
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### How to navigate around a smooth surface?

Suppose I want to find the shortest path between two points in $\Bbb{R}^3$ with smooth obstacles in the way? I understand things like Dijkstra's algorithm for shortest paths on a graph. But what about ...
23 views

### Maximum / Minimum Question with 3 Variables?

I seem to be stuck in this problem, would need your help! Question: Assume I have : 147 of x, 174 of y, 238 of z A different amount of x, y ...
15 views

### Calculating Normals across a sphere with a wave-like vertex shader

This is a bit of a CS question, but more than not it's a 3D math problem. I've been trying to get the correct normals for a sphere I'm messing with using a vertex shader. The algorithm can be boiled ...
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### If $A,B$ are proper subsets of connected space $X,Y$, then the set $(X\times Y)\setminus (A\times B)$ is connected

Let $A$ be a proper subset of $X$ and let $B$ be a proper subset of $Y$. Prove that $(X\times Y)\setminus (A\times B)$ is connected whenever $X$ and $Y$ are connected?