Server or "Workstation"
4 fast cores or 6 slower ones?
find the number of injections/surjections I've been banging my head on this problem for some time now, and could really use help. Bear in mind I'm not very good at this sort of thing and am struggling to get by in class.  Problem:     Given $ \alpha : A \to B       \;\;\;\;\;   |A|=p \;\;\;\;|B|=q $      Find:         Number of Injections: $A \to B$   Number of Surjections: $ A \to B$    Prove that ∃ a bijection $A \to B$       $\Rightarrow$ p = q   find number of bijections: $A \to B$         I apologize for dropping such a big problem on you all, but I just can't get anywhere with this. Help?  EDIT: These answers are really helpful and I believe I've got down the number of Injections, but  we haven't gone through permutations or combinatorics in class so my understanding of it is really hamstrung. I'm unclear on the Stirling theorem as discussed here: , specifically the part where inclusion-exclusion comes into play. I'm not sure how to apply that, since right now I'm letting p = 4 and q = 3, so I don't know where inclusion-exclusion would be used (if at all!) since I don't know how elements would be missing. Why does the zoom sometimes stop at a point? What is a pull up and pull down? Can some one explain this terminology please I think I understand it but not completely sure.   I think pull down is where you place a resistor between +V and the other component and pull up is where you place the resistor between 0v and the component.    If I am completely wrong then let me know! How many solutions for an equation with simple restrictions I'm working on an assignment in which I have to count the number of solutions for this particular equation: $$x_1+x_2+x_3+x_4=20$$for non negative integers with $x_1&lt;8 $ and $x_2&lt;6$   I'm aware that this kind of a task isn't that complicated, but I don't get combinatorics in general that well.  So I've tried two following approaches to get this done.  Firstly I tried to substitute the variable x:  $x_1+x_2+x_3+x_4=20 \Leftrightarrow y_1+y_2+y_3+y_4=34$  in which $y_1=x_1+8$ and $y_2=x_2+6$ (casue $x_1=y_1-8$ and $x_2=y_1-6$) Following this approach the total number of possible solutions would be  $${34+3 \choose 3} $$  But I'm not sure if its the right solution.  The second approach is to sum all of the possible values that $x_1$ and $x_2$ could possibly take, also $x_1=0,1,2,3,4,5,6,7$ and $x_2=0,1,2,3,4,5,6$ And then  count all the possibilities for each of the variables $${20 -x_1-x_2+1\choose 1}$$ and sum them together like this: $${21\choose 1}+{20\choose 1}+{19\choose 1}+{18\choose 1}+... $$ and so on...  I'm sure I'll get the correct number with this one, but I'm not feeling like summing all of this possibilities. There's got to be a better, more elegant way to deal with this.  My professor gave me a hint that I should do it using the complement. I was asked this question during an interview for a trading position with a proprietary trading firm. I would very much like to know the answer to this question and the intuition behind it.  Amoeba Question:  A population of amoebas starts with 1. After 1 period that amoeba can divide into 1, 2, 3, or 0 (it can die) with equal probability. What is the probability that the entire population dies out eventually? Filtering questions by multiple tags I know you can select a tag and then click on unanswered to view the unanswered questions for that tag.  But is there a way to view questions only for a certain set of tags > 1 tag?  Mainly I'd like the union of tag1 with tag2 with tag3 with tag4. But also the intersection of tags would be nice as well. Site being repeatedly hacked: any useful tips for tracking down entry point? Where were the capital ships and bombers? Ch.3 #7. Prove that the convergence of $\sum a_n$ implies the convergence of $$\sum \frac{\sqrt{a_n}}{n},$$ if $a_n \geq 0$.      My attempt. If $\sum a_n$ is convergent, then by the root test, $\lim_{n \rightarrow \infty} sup \sqrt[n]{|a_n|} &lt; 1$. This means $|a_n|^{1/n} &lt; 1$. Because $a_n \geq 0$, this means $a_n ^{1/n} &lt; 1$. Now apply the root test to $\sum \frac{\sqrt{a_n}}{n}$ by considering $$\lim_{n \to \infty} sup \sqrt[n]{|\frac{\sqrt{a_n}}{n}|}.$$ $$\sqrt[n]{|\frac{\sqrt{a_n}}{n}|} = \frac{\sqrt{{a_n}^{\frac{1}{n}}}}{\sqrt[n]n}.$$ Because  $a_n ^{1/n} &lt; 1$, certainly the numerator on the left-hand side of the equation above is $&lt;1$. And because $n$ tends to $\infty$, and $n &gt; 1 \rightarrow \sqrt[n]n &gt; 1,$ the denominator is $&gt;1$. Therefore the limit is $&lt;1$ and the root test implies that $\sum \frac{\sqrt{a_n}}{n}$ converges.     If there are any errors in my proof, I think they come from 1) my assumption of the converse of the root test theorem and 2) my misunderstanding of $\lim_{n\to\infty} sup$. Thanks in advance for your help. Why is there such a big difference between "Size" and "Size on disk"? Google Image Search using Android I have an image on my smartphone and I want to paste it into Google Image Search to see if I can get similar images.  I can do this by pasting the image using desktop but not using Android ...  Is there such a possibility? Extract all string from a java project I recently found a formula to help find a sequence given a generating function:  $(\frac{1}{1-x})^k=\sum_{n=0}^{\infty}\binom{n+k-1}{n}x^n, k \in \mathbb{N}$  I just wondered if there was a known proof for this, or if anyone knows of a way to prove it. In Python, I can compile a regular expression to be case-insensitive using re.compile:  &gt;&gt;&gt; s = 'TeSt' &gt;&gt;&gt; casesensitive = re.compile('test') &gt;&gt;&gt; ignorecase = re.compile('test', re.IGNORECASE) &gt;&gt;&gt;  &gt;&gt;&gt; print casesensitive.match(s) None &gt;&gt;&gt; print ignorecase.match(s) &lt;_sre.SRE_Match object at 0x02F0B608&gt;   Is there a way to do the same, but without using re.compile. I can't find anything like Perl's i suffix (e.g. m/test/i) in the documentation. How to delete a file in Windows with a too long filename? My wife has several files and folders that somehow ended up having filenames that have caused them to be undeleteable (can't be deleted) by normal means or via the command line. I believe the filenames are too long due to the depth of the folder structures. Does anyone know of a good utility for cleaning up files like this? Writing a Virtual Printer in .NET Do sets, whose power sets have the same cardinality, have the same cardinality? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. Closing Reject Suggested Edit dialogue freezes review process Why do we need Diffie Hellman? Diffie–Hellman offers secure key exchange only if sides are authenticated. For authentication, sides are using public/private key. So if side A knows the public key of the side B, then A can simply use that key to encrypt the message (or the new session key).  Why bother to use separate key exchange algorithm? Can there be an option to allocate rep from any site account for offering a bounty? I think it would be useful if we had an option when deciding to open a bounty on a question that we could choose to allocate rep from one site even though the bounty is being offered on another site.  For instance I would like to use my rep from SO for putting a bounty on a question that resides on SF. I don't have enough rep on SF to make it enticing. So when you click on "start a bounty" you get sliders for each site that you can pull your rep from, maybe make it where you need to burn up all your rep for the site that your asking the question on and then allocate more rep from other ones, if you have it.  thanks devinb for clearing up my question with your comments! " transfer rep INTO a bounty. But you can't transfer rep across accounts. So, the rep you'd 'lose' from SO would never actually go into your SF account, it would go straight onto a bounty on whatever question." – devinb