Prove that,  $$\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$$  Here is my solution below, this is my first time working through a pascal identity problem and I just wanted to see if this made sense:  By the definition $\binom{n}{r}$ we have,  $$\binom{3n-1}{r-1}=\frac{(3n-1)!}{(3n-1-(r-1))!(r-1)!} = \frac{(3n-1)!}{(3n-r)!(r-1)!}$$  and,  $$\binom{3n-1}{r}=\frac{(3n-1)!}{(3n-1-r)!r!}$$  Thus, starting with the right hand side of the equation,  $$\binom{3n-1}{r-1} + \binom{3n-1}{r} = \frac{(3n-1)!}{(3n-r)!(r-1)!} + \frac{(3n-1)!}{(3n-1-r)!r!} \\ = \frac{(3n-1)!r}{(3n-r)r!} + \frac{(3n-1)!(3n-r)}{(3n-r)!r!} \\ = \frac{(3n-1)!(r+3n-r)}{(3n-r)!r!} \\ = \frac{3n!}{(3n-r)!r!} \\ = \binom{3n}{r}$$ \
Reading through my textbook I came across the following problem and I am looking for some help solving it.  Prove that,  $$\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$$  In a previous problem I solved the following but I wasn't sure how to use it to prove the question above. Prove,  $$\binom{n}{r}=\binom{n}{n-r}$$  Solution:  left Side,  $$\binom{n}{r}=\frac{n!}{(n-r)!r!}$$  right side,  $$\binom{n}{n-r}=\frac{n!}{(n-(n-r))!(n-r)!} = \frac{n!}{(n-r)!r!}$$  Therefore completing the proof,  but how do I solve my first question? thanks!
Where do I get the key for the first house? I've 'completed' the game, but didn't find the key for the first house. (I got the one for the arcade machine house). Where can I find it? Can an Indian citizen holding a valid USA visa visit Colombia without a visa? I am planning to travel to Colombia for about a week to attend an academic conference. I am an Indian passport holder currently living in USA with a F-1 visa. Do I need to obtain a visa beforehand from the Colombian consulate?  I have found   that claim that Indian citizens holding a valid USA visa are not required to apply for Colombian visa.     However, I am not sure if these sources are "official" and trustworthy.Is there a more official source? Also, does this rule apply to "tourists" only? Does anyone have any personal experience with this rule? Number of lines in a file in Java I use huge data files, sometimes I only need to know the number of lines in these files, usually I open them up and read them line by line until I reach the end of the file  I was wondering if there is a smarter way to do that How to export data files using specific number format Would Beetee's electrocution plan work and kill the tributes in the lake? In , the second episode of the Hunger Games trilogy, one of the characters (Beetee) proposes a plan to kill some of the remaining tributes. He proposes wrapping a special wire that he has designed to withstand the vast energy of lightning around a tall tree and running it down to a salt water lake. Then when lightening strikes the tree (which it does predictably every twelve hours) anyone in the lake will be killed. (N.B. This is not Beetee's real plan, but it is the one he wants his allies to believe.)  I don't think his plan would work at all. Assuming that his wire can carry the enormous current delivered by a lightening strike (this is fiction after all) surely the current would short to ground through the salt water rather than taking a path through anyone standing or swimming in the water. The analogy I have in mind is that of birds standing untroubled on live electricity wires. They are unharmed because they do not provide a path to earth. But another analogy might be dropping a mains electric heater into a bath when you are in it. I think that kills you (though I'm not sure why.) Flying Turkish Airlines, what are credit card rules? I have return tickets flying out of SFO on Turkish airlines for the whole family. The credit card I used to book the tickets has since been replaced by the bank with a new card.   In order to board, can I present the new credit card (same name) at the checkin counter, or must I present the original credit card (which is no longer valid and has been discarded)?  By 'discarded' - I mean cut up and thrown away in trash. It was no longer valid, since it was replaced with a new card. So can I use the new card to travel on a booking that was made using the original card?    The new card has a new number; and the ticket was purchased through an agent. All html canvas shapes are given the color of the last object added Events triggered by dynamically generated element are not captured by event handler I work sometimes till 4 am and my friends and the rest of the world obviously don't know in advance that I don't want to be called at 8 am.  Is there an app or setting that can allow me to have an emergency override list of people I need to be woken up for if a possible emergency arises but ignore all other calls until I wake up?    I'm familiar with the Do Not Disturb option, but it has limitations that it blocks all calls. What if there is an emergency at night?  What if I want to allow certain callers during the Do not Disturb time? Is there a way to have an emergency list that allows calls during silent? Is there way to move one vertex and make another move with it proportionally? I'll try to explain in this gif. I am doing this manually in 2 steps - rotate, scale. And I select all vertexes. Is there instrument rotate them in one step or can I move one and make other move too? Is a DSL splitter required, if only a Modem is attached to the line? Imaging new hard drive in Windows 7 laptop? I'm upgrading my Windows 7 laptop to a new hard drive.  How can I clone or image the old hard drive over to the new hard drive, so that it looks like nothing has changed (other than the new hard drive) to my operating system and applications?  Can I use what is built into Windows 7 Backup or should I use something else? set fontsize for TikZ figure Programmatically open new pages on Tabs How to remove garlic smell from refrigerator/ice maker Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$? Is it possible for a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$,  $$\mathbf{A}^2=-\mathbf{I}$$  I know that It is possible for $2\times2$ matrix, but is it possible for $3\times3$ matrix ? What makes a good jump scare? I am working on a game and would like to include a few jump scares. So here is a rather simple question — What makes a good jump scare and when and how should they happen? How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? I'm building a Core  system which supports dual-channel memory (up to 16&nbsp;GB in four DIMMs), my target is 8&nbsp;GB. From the performance perspective, is it better to have two times 4&nbsp;GB or four times 2&nbsp;GB? Intuitive explanation of binomial coefficient formula