How to solve Frog-stair problem with combinatorics instead of recursion?
How many ways can $b$ balls be distributed in $c$ containers with no more than $n$ balls in any given container?
Expression for a choice which isn't really one GPG can't check signature $n$ choose $k$ where $n$ is less than $k$ As someone who doesn't play Team Fortress 2, I occasionally get people asking me to trade, but I have no idea what's a "fair" price for these items. I don't really get this whole trading thing (I've seen people ask for "63 buds", I have zero idea what that means), but I don't want to get ripped off.   Some recent  examples would be:        How can I figure out how much an item is worth? What "lingo" do I need to know in order to trade? Mathematical Basis behind inflation of Standard errors of Regression estimates due to multicollinearity We know that due to multi-collinearity, the standard errors of beta estimates get inflated. But what is the mathematical basis to it?   I am looking for some mathematical relationship or something to explain this.   Like I understand if standard error of betas goes up, then t-statistics goes down and we might not be able to reject the null or the variables would appear non-significant.   But what is the mathematical relationship between multicolllinearity and inflation in variance of coefficients? When you create a script via the Unity editor, it generates a script with some pre-formatted code.  using System.Collections; using System.Collections.Generic; using UnityEngine;  public class GenericClass : MonoBehaviour {      // Use this for initialization     void Start () {      }      // Update is called once per frame     void Update () {      } }   When I create a script, I am generally guaranteed to use additional code, such as a namespace or a custom editor. Furthermore, I almost always delete content from the automatically-generated script. Is there a way to change the automatic code generated by Unity? How do I study the "correlation" between a continuous variable and a categorical variable? What is vertex and edge shading? Why $SU(3)$ and not $U(3)$? If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK. Note K is a normal subgroup of HK  I thought I could just prove K is a subset of HK and vice versa so as to prove it a normal subgroup but then how to do this I have no idea because I keep getting hk element again while trying to do so.  I just tried to implement the steps to find HK is a subgroup of G. Is it safe to use the hash of the data as the key to encrypt them? Is it safe to use the hash of the data as the key to encrypt them?  That is, if I give one some encrypted data and tell them that the key that was used to encrypt them was the hash digest of the data before encrypting, will it be easier for them to decrypt them?  Edit as I can not reply to comments: The recipient will have the keys, that is no problem. (a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture. (b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical induction to prove your conjecture  I understand that for part a that all multiples of 3 of n are even. So F(0),F(3),F(6)... I just don't understand how to prove it.  For part B it is the same thing except multiples of 4  Please help, thank you! I'm trying to convert some LaTeX into HTML but my code listings don't preserve their code coloring. Why is that and how do I fix it?  This is what it looks like:    Side note: I'm also wondering why the caption on the table was removed.  Edit  Here is an example listing with colors:  \documentclass{article} \usepackage{color} \usepackage{listings} \lstset{     basicstyle=\footnotesize\ttfamily,     language=[Sharp]C,      keywordstyle=\color[rgb]{0,0,1.0}\bfseries,       commentstyle=\color[rgb]{0.133,0.545,0.133}\bfseries, }  \begin{document} \begin{lstlisting} int x = 10;  // This is a comment string str = "hello"; \end{lstlisting} \end{document}   This produces the following HTML when doing: htlatex myfile.tex:  &lt;!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"     "http://www.w3.org/TR/html4/loose.dtd"&gt;   &lt;html &gt;  &lt;head&gt;&lt;title&gt;&lt;/title&gt;  &lt;meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"&gt;  &lt;meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)"&gt;  &lt;meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)"&gt;  &lt;!-- html --&gt;  &lt;meta name="src" content="LaTeX4.tex"&gt;  &lt;meta name="date" content="2012-07-20 12:30:00"&gt;  &lt;link rel="stylesheet" type="text/css" href="LaTeX4.css"&gt;  &lt;/head&gt;&lt;body  &gt;    &lt;!--l. 12--&gt;    &lt;div class="lstlisting" id="listing-1"&gt;&lt;span class="label"&gt;&lt;a   id="x1-2r1"&gt;&lt;/a&gt;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;int&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;x&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;=&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;10;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;br /&gt;&lt;span class="label"&gt;&lt;a   id="x1-3r2"&gt;&lt;/a&gt;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;br /&gt;&lt;span class="label"&gt;&lt;a   id="x1-4r3"&gt;&lt;/a&gt;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;//&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;This&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;is&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;a&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;comment&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;br /&gt;&lt;span class="label"&gt;&lt;a   id="x1-5r4"&gt;&lt;/a&gt;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;string&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;str&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;=&lt;/span&gt;&lt;span  class="cmtt-8"&gt;&amp;#x00A0;&lt;/span&gt;&lt;span  class="cmtt-8"&gt;"&lt;/span&gt;&lt;span  class="cmtt-8"&gt;hello&lt;/span&gt;&lt;span  class="cmtt-8"&gt;"&lt;/span&gt;&lt;span  class="cmtt-8"&gt;;&lt;/span&gt;     &lt;/div&gt;  &lt;/body&gt;&lt;/html&gt;   Which looks like this:    I would want the HTML result to look like this: Bizarre spacing with 4 $\star$s The spacing between the 3rd and 4th stars is non-existent. Anyone know what's up?    \documentclass[11pt]{article}  \usepackage{amsmath}  \begin{document}  \begin{gather*} \star\star\star\star \\ \star\star\star\,\star \\ \star\star\star\star\star \end{gather*}  \end{document} If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: $$(r+I)\cdot m=rm.$$ (a) Show that this product is well defined. (b) Show that this product, together with the sum of elements of $M$, turns $M$ into an $R/I$-module.  My answer for (a): Let $r+I,s+I \in R/I$ such that $r+I=s+I$. Then $(r+I)m=(s+I)m$, for all $m \in M$. Then $rm=sm \implies r=s$  (since the inverse of $m$ is in $M$), so the product is well defined.  Is my answer to (a) correct?  How can I use Ann(M) to answer (b)? LINQ to Entities does not recognize the method 'System.String Format(System.String, System.Object, System.Object)' Filter to change the content of 404 page I have created custom rewrite rules. These rules now go to 404 page as expected. I can use action hook template_redirect to change the template matching query_var.  But I don't want to create separate template files as the page structure depend on the current theme in use. Rather I just want to change the content and title. I have tried change the content using the_content filter, but it seems the filter doesn't even get called for 404 page.  I also tried loading index.php template and then using the_content filter, but it also doesn't work.  Is there any way to achieve this? What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks Date Formatting C# Where do the American and Japanese Transformers franchises meet? The original Transformers TV series was created by the American company Hasbro, and was adapted into Japanese as well. At some point, the Japanese franchise split off into its own continuity, launching its own set of sequels and spin offs.  The questions is: What part of the American franchise is considered canon by the Japanese one? Minor details such as name changes are not to be taken into consideration.