Prove that the number of permutations of $m$ A's and at most $n$ B's equals $\dbinom{m+n+1}{m+1}$.  I'm not sure how to even start this problem.
I have to prove that:     $$\sum_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$$   I tried to open up the right side with Pascal's definition that: $$ { n \choose k} = {n-1 \choose {k}} + {n-1 \choose {k-1}}$$  Here is what I came up with, and I am sure it is wrong because it does not equal the left side:  $$ {m+n+1 \choose m+1} = {m+n \choose m+1} + {m+n \choose m} = ... ={m+n \choose m+1} + {m+n-1 \choose m} + {m+n-2 \choose m-1} + ... + {m \choose m+1} = \sum_{k=0}^{n} {m+k \choose m+k-n+1 }  $$  Which, again, probably is wrong because it is not equal $\sum_{k=0}^{n} { m+k \choose m}$. Any help is appreciated
Antonym of "phobia" If a phobia is to have an irrational fear of something, what is the word for having an irrational affinity for something?  For example a numerologist may fear the number 13, but be attracted to (or even have a love for) the number 8. (Some would say this is irrational.) Proper capitalization of commonly used acronyms and initialisms I was always under impression, that acronyms ought to be written all caps. However reading  site very often I see some of the common acronyms written as proper names (first cap). For example "Nato" instead of "NATO".   I know that BBC is quite strict in proper English usage, so what would be the reason behind this? How can I take a screenshot with Selenium WebDriver? Careers Stack Overflow - Tag "ruby-on-rails" returns jobs that don't contain the tag Is logic subjective? If logic is constructed from axioms, and axioms are depended on observation which in term could be subjective, does this means that logic could be limited to our observation, and not really absolute and fundamental? Stacks for Unity Launcher I'm looking forward to emulate the behavior of OS X Dock Stacks so that I can have, for instance, one stack for browsers and another one for IDES. People at ask ubuntu are doing a lot of cleaver stuff with custom Launchers and Quicklists (see ). So has anyone figured out how to have stacks in Unity? Anything fancier than a static Quicklist over a dummy folder icon? How to move a visible file to a hidden folder? I've only been using Ubuntu for two weeks and have been able to find answers to any problem I had, however this one defies my searches.  I want to move a visible programme file (wallch) to a hidden directory ./config/autostart.  I have been able to enter this directory and the subject file is not there and I want to have this program start at bootup.  Can anyone assist?  Thanking you My friend, John, and I went shopping.    My friend John and I went shopping.       I believe we need to wrap our friend's name in commas (first sentence example), but wouldn't that make it three people? Dynamic Table partitioning on a daily basis I have a SQL Server database which contains two tables -- Acks and Logs.  These two tables are related in logic but not in a relational database way. Basically, every message that comes in gets saved in the Log table and, if our server acknowledges it, then that ack gets stored in the Ack table.  We are storing around 5 million Acks and 3 million Logs a day. I am trying to partition these two tables on a daily boundary so that we can easily remove older partitions from the table, and also improve query performance.  I haven't done table partitioning before, so I have been reading some on-line tutorials, however I am stuck on one thing. All the tutorials I have followed seem to manually add filegroups and manually add boundaries.  I want SQL Server somehow to do this daily, and this is what my question is about. I need it to create the new filegroups for the next day, every day at, say, 22:00. Then at 24:00 the inserts should start filling up the new day's partition.  Can anyone point me in the right direction on how to achieve this? Either a comprehensive tutorial or some good old advice will do as well.  My second question: can I somehow apply the same partition function to the two different tables?  They both have a datetime(2) column on which I want to partition, and the same rules  will apply.  How does that then fit in with my filegroups? Do I need a single filegroup for the day? Will each table have a file in that filegroup, or will both tables save to the same file in the filegroup?  Do I have to make a .mdf and .ldf for each filegroup? Or is there still one log file for the entire database? Doubt related to a theorem in congruences of Chapter 5 of Apostol Introduction to analytic number theory I am self studying number theory from Introduction to Analytic number theory by Apostol.   I have a doubt in one argument of proof of Theorem 5.14 and I am posting its image.      My doubt is in 2nd line of 2 nd paragraph how Apostol writes - If y is a solution of (5) then ay $\equiv $ at ( mod m) . How does y being a solution of (5) implies that?       Can someone please help ionice without effect How do I identify SMD components? (or how do I identify any component) How do I identify the markings on an SMT component and match it up with a part number so I can be a good designer and actually look at a datasheet (and read the whole thing)? Or identify a part to replace an unknown part on a PCB? Quotient ring of Gaussian integers What happens when a caster targets an object that looks like a creature with a spell that targets only creatures? There are many spells which explicitly target creatures:     Choose one creature within range   The target creature is normally supposed to make a saving throw in this case.  What happens when the caster chooses a creature, that is actually an object? For instance, a scarecrow, a mannequin, an illusion, or a corpse.  Examples:   An evil mage is hiding in the bushes. He casts an illusion of himself on the road, using the Silent Image spell. A PC sorcerer walks down the road, sees the illusion and casts Magic Missile on it. A cleric is standing near a pile of dead bodies. He has suspicions that a few corpses are actually undead creatures. He casts Sacred Flame on every corpse, see which if them succeed, hence, reveals the undead. A wizard casts Acid Splash on a mannequin, trying to damage it with acid.   Possible outcomes I can think of are:   The caster cannot even try to cast the spell, regardless of their information about what the target actually is. The caster tries to cast the spell, the casting fails, not expending the spell slot. The caster casts the spell, it expends the spell slot, then fails. The caster casts the spell, but it hits the original and not the illusion. The caster casts the spell, it flies to the target illusion but deals no damage to it (passes through it).   References in the rules and Sage advice  I was trying to investigate, but it didn't make things clearer:    has an answer, that Magic Missile won't hit. It says nothing about if it can be targeted at all, it leaves it up to the DM. Magic Missile automatically finds the original, even when the caster targets an illusion. At least . Now . This supports the counter-intuitive option 4.   on the other hand   , if there were no perceivable effects (let's say you can try to cast the Command spell on an illusion). That assumes that casters can waste their spells on illusion, which means they at least can target it. This supports options 3 and 5. 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? Why does comparing strings using either '==' or 'is' sometimes produce a different result? How to change sign text color in Minecraft? I have gone through every single website and video to try to make a sign with colored text, but nothing works. Can anyone help me to get a sign to say:  Place Block(dark blue and bold text)  I really want this for a map I am making and I need help. In the United States, we say that someone lives on a street, whereas I've noticed that British people say in. For instance:     Bubba lives on Washington Street.   Colin lives in Cavendish Avenue.   I believe we both would use at when a number is given. For instance:     Bubba lives at 16 Washington Street.   Colin lives at 7 Cavendish Avenue.   I don't think it matters if it's a road, avenue, street, circle, or lane — as far as I know, in America, we always live on a street. Does it vary in British English, or is it always in? If it varies, what are the rules? Mark my own question as favorite? Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"?