Assume we toss a coin 10 times, independent of each other.  Each time we can get Heads (H) or Tails (T) , regardless of whether it is fair or not.  So for example this is one possible outcome: HHHHHHHHHH ie 10 heads. Let's call this a 10-toss sequence.   The total number of possible 10-toss sequences is 2^10 because each toss has 2 possible outcomes: Heads or Tails.  --The first question is What is the meaning of 10!/(3!7!) ? The answer is that It is the total number of ways, by which we can place the three heads inside the 10-toss sequence. The order by which we place the three heads does not matter.   --The second question is : What is the total number of possible 3-head sequences? Is it 10!/(3!7!)? Some say YES. Others, say NO. A 10-toss sequence with three heads, can be this one: HHH THTHTHT. So three positions are fixed to 'heads'. The remaining seven positions can be H or T. So we have 2^7 possible 7-toss sequences. And the three heads can be anywhere in this 10-toss sequence.   So an answer can be that the total number of 10-toss sequences with 3 heads is 10!/(3!7!) multiplied by (2^7) . That is in the 10-toss sequence the number of ways by which we can place 3 heads is 10!/(3!7!). Then, we have to say something about the remaining 7-toss sequence. Each position can take Heads or Tails . So 2^7 is the possible number of this 7-toss sequence.   However, I know that the correct answer is 10!/(3!7!) and that we don't need to multiply by 2^7. But I cannot understand why.
This is a question that has received various answers. This questions seems simple, but it proves to be rather tricky.  So :  We flip a coin (H-T) independently $10$ times. Let's for the moment not care about whether it is fair or not.   How many $3$-head sequences are there?  Solution:  Some say that the answer is $\displaystyle\frac{10!}{7!3!}$, but this is the number of ways  by which $3$ heads can be arranged in $10$ positions. This answer , however, does not say anything about the remaining $7$ positions.   Each of these seven positions can be H or T. The total number of $7$-element sequences is $1\cdot1\cdot1\cdot2^7=2^7$. Then we have to multiply this by $\displaystyle\frac{10!}{7!3!}$ as the three heads can be anywhere. But I have found that this is not the correct answer. The correct is just $\displaystyle\frac{10!}{7!3!}$. So why not multiply by $2^7$?
Short brandable domain vs. long keyword domain I need a counsel about my websites' domain.  Let's say I have just bought two domains: xyz.com and xyzphotography.com.  As we can see, the first one is shorter but way less meaningful than the second one. The second one contains a photography keyword but it's longer.  I have only one website which is my photography portfolio. Can you advise me how to set those domains to make it all as SEO-friendly as possible? Which one should be the main one, does the rest should work on their own or have some kind of SEO-friendly redirect to the main domain? Create unmanaged c++ object in c# How can I ensure my trip to the UK will not have to be cancelled because of Brexit? I've got a trip planned to the UK in June. Due to brexit, I'm worried what changes may be introduced before my arrival.  I'm traveling on a EU / Polish passport. I'm going for a couple weeks of holiday.  Is there anything I can do right now to make it less likely any changes won't stop me at the immigration? For example, applying for some tourist visa which does not rely on the EU status? Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$ Where to start looking up if there's something wrong with cck field variables? in short: Something in my zen-based custom theme is keeping a couple of cck fields from being display on my page. Basically my question is where to get access to the variables that are supposed to contain the html for the cck fields (which are missing when I was looking them in firebug). Below I explain why I think it's looking up the variables that might bring me closer to a solution  So I made this custom node type (events) with cck fields added. Also made some content using those fields. Added some css and it all displayed fine. Later the taxonomy terms below the title and some metadata bugged me so I followed along with a 'theming cookbook' pdf book that taught me how to make a node-events.tpl.php and simply deleted php variables like $terms. This worked great.     Obviously this also means that all this time, the view permissions of those fields have been checked and also that the display options under 'display fields' have been set to standard all the time. I also ruled out that the issue could be anything else than theme related as my cck fields do show up in a stock Drupal theme. The theme for my site is a zen-based custom theme.   So then even later, after I made a view for this node type and started adding css to that view I noticed at some point that my image field and two other fields are missing. It's hard to tell if they disappeared because of things that I did when creating the view and adding css (which would surprise me) Limit Decimal Places in Android EditText X forwarding from Ubuntu Server Ed to Windows 7 The Goal: I would like to open a program from my ubuntu server in my windows 7 desktop.  Steps Taken: Putty > SSH > X11 check 'enable port forwarding' I have started VNC Viewer in listening mode started putty ssh session apt get install xinit startx &amp; geany (it's a c language ide; it uses the GUI so I"m using it to test x forwarding)    Error Messages: geany returns "cannot open display" startx returns "setversion 1.4 failed"    I'm working on Windows 7 and headless Ubuntu Server 12.04 LTS  Obviously I forgot something, but I'm having difficulty figuring out what. The goal is to not use sudo apt get install desktop, since I only want to use one program at a time, and anyway my server isn't hooked up to a monitor.   Any ideas? Questions? Thanks for your help! Curios Running macOS Sierra(Version 10.12.2 16C67).  I installed  version wine-1.8.6 (through a .pkg file). I opened up a game and it worked fine.  The next day when I downloaded Origin installer, it wouldn't open. When I double clicked on it, the Wine icon showed up in the dock and said it was open, but no windows showed up. When I clicked on another program(or on the desktop), Wine closed. Now when I try to start that same game I opened when I first installed Wine, the same thing happens that happened when I started Origin installer.  When I go to the terminal and type wine {path of .exe} I got this:   I installed wine tricks using brew install winetricks and it installed fine, yet it didn't change my situation.  winecfg works properly, and so does sh winetricks. Why is $ 561 = 3*11*17 $ the smallest Carmichael number? A composite integer n is a Carmichael number if the only Fermat   witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not   coprime with $n$. The smallest example of such a number is $561 = 3 ·  11 · 17$   I think I understand the definition, but I don't quite understand why is the smallest Carmichael number is the above mentioned number. Please clarify. I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) $3$-cycle is a set of $3$ (distinct) vertices, $a, b, c$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.  I know there are similar ones already on here, but I can't comprehend any of the answers. This is what I have so far:  Base Case: Since the graph has to have at least $3$ vertices to have a simple $3$-cycle, the lowest possible even number of vertices is $n = 4$.   $\dfrac42 + 1 = 3$ degree (edges attached to node). This means each vertex connects to each of the other $3$ vertices since $G$ is a simple graph. Thus, any combination of $3$ vertices will be a simple $3$-cycle.  Inductive Hypothesis: Assume a simple graph with $k$ vertices, where $k$ is even, contains a simple $3$-cycle if each vertex has degree $\dfrac k2 + 1$.  Inductive Step: Since only concerned about when the graph has an even number of vertices, going to show a simple graph with $k + 2$ vertices, where $k$ is even, contains a simple $3$-cycle if every vertex has degree $\dfrac{k+2}{2} + 1$   This is as far as I get and don't know how to finish the Inductive Step (Also please feel free to let me know if I messed anything up, up to this point) Why does one get an illusion as moon following him? When you run or ride bike at night if you observe the moon you feel like he moves along with you as the same speed you are going. Why? Algorithm to find Largest prime factor of a number What is the best approach to calculating the largest prime factor of a number?  I'm thinking the most efficient would be the following:   Find lowest prime number that divides cleanly Check if result of division is prime If not, find next lowest Go to 2.   I'm basing this assumption on it being easier to calculate the small prime factors. Is this about right? What other approaches should I look into?  Edit: I've now realised that my approach is futile if there are more than 2 prime factors in play, since step 2 fails when the result is a product of two other primes, therefore a recursive algorithm is needed.  Edit again: And now I've realised that this does still work, because the last found prime number has to be the highest one, therefore any further testing of the non-prime result from step 2 would result in a smaller prime. I am fascinated by the way the CLR and GC works (I'm working on expanding my knowledge on this by reading CLR via C#, Jon Skeet's books/posts, and more).  Anyway, what is the difference between saying:  MyClass myclass = new MyClass(); myclass = null;   Or, by making MyClass implement IDisposable and a destructor and calling Dispose()?  Also, if I have a code block with a using statement (eg below), if I step through the code and exit the using block, is the object disposed of then or when a garbage collection occurs? What would happen if I call Dispose() in the using block anyay?  using (MyDisposableObj mydispobj = new MyDisposableObj()) {  }   Stream classes (eg BinaryWriter) have a Finalize method? Why would I want to use that? Execute shell commands in Python Do Cleric's domain spells add slots? I'm a little bit confused with the whole spell system and additions to them. If I pick for instance Tempest Domain Spell, are the spells additional to the other ones? (2 at level 1) So do that make it 3 spells? Does the domain add slots too?  And also, I've read about people choosing multiple domains, is that possible? Story ID: A pre-teen girl who is a genius crosses the country in a van How to disable Chrome's Incognito Mode? How to adjust the gap between figure and caption within the same subfloat How to install Google Chrome If A is a finite set and B is an infinite set, then B\A is an infinite set Prove: If $A$ is a finite set and $B$ is an infinite set, then $B \setminus A$ is an infinite set.   I've tried proving the contrapositive, that if $B \setminus A$ is a finite set then $A$ is infinite or $B$ is finite, but I don't think this is the way to go.