Before I go diving into the book could I tell you what an old teacher of mine used to say about "Some people think "n" is not a number. but then how can n = 3?"
@FreakEnum OK. First, I will state the question clearly for reference. There are 6 boys and 6 girls, they are made to sit in a line in a random order. What's the probability no two girls end up next to each other?
Sample space = 12!; now, Lets 6 boys sits first on 6 chairs , then no of ways = 6!; now girls can be sit on 7 places i.e Boy= "*" , Girl="#" , Seat = "_" , so _*_*_*_*_*_*_ so girls can be sit in = 7P6 ways;
@DylanMoreland Actually, this is the not the first user with such a name. There was a someone with a handle like "qqqqq*qq*qqqqqqq" (The *'s were actually two other letters. I don't remember the exact handle, obviously.)
@QED For what it's worth, I disagree with this viewpoint. Actually, it does make a lot of sense, provided it works. However, there are many situations where you see things only hazily -- where one is forced to guess, to stumble, to err. It may not happen for such textbook problems, but I think it's inevitable when you start doing more and more nontrivial ones, let alone in research.
To be honest I don't really believe that math research is possible seeing how many centuries people have been working on it (but I'm obviously wrong because people are doing it all the time)
A timeline of pure and applied mathematics history.
Before 1000 BC
* ca. 70,000 BC — South Africa, ochre rocks adorned with scratched geometric patterns.
* ca. 35,000 BC to 20,000 BC — Africa and France, earliest known prehistoric attempts to quantify time.
* c. 20,000 BC — Nile Valley, Ishango Bone: possibly the earliest reference to prime numbers and Egyptian multiplication.
* c. 3400 BC — Mesopotamia, the Sumerians invent the first numeral system, and a system of weights and measures.
* c. 3100 BC — Egypt, earliest known decimal system allows indefinite counting by way of introducing n...
The friend who hooked me up has disappeared off the face of the Earth, kicked out by his wife, and the place said they have no idea who he is. And he took some of my shit, which really pisses me off.
Hmm, well, if you sum $n^{-s}$ over squarefree integers equal to $a\mod b$ you get $$\sum_{\chi\mod b}\frac{\overline{\chi}(a)}{\varphi(b)}\frac{L(s,\chi)}{L(2s,\chi)},$$ that's something at least.
But if I must make a guess, I would do the following. For simplicity, let's first assume $b$ is a prime number, and $a$ is nonzero modulo $b$. Then $n = a \pmod b$ is square free if for every prime $p \neq b$, $p^2$ does not divide $n$. There is no condition imposed by $b$. So this gives the probability $$ \prod_{p, p \ne b} \left( 1 - \frac{1}{p^2} \right) = \frac{6}{\pi^2} \cdot \frac{b^2}{b^2-1}. $$
@robjohn, can you explain to me what the following excerpt is trying to say: Hardy's "Course..." I'd need to investigate years-of-publication and such, but I know the provenance of that thing needlessly well. My father (a high school math teacher in the U.S. required to obtain a higher degree [sic] in mathematics) endured a night-school course whose text was Hardy's. I was not very old, but old enough to be ... stunned... by the tendentiousness of that text. I had a fair understanding of analysis in the late 1960s, and/but Hardy's text effectively expressed doubt that its reader "had a brai…
I am especially stuck at the word "tendentiousness". Dictionary search says something like "having a tendency", "biased". But that doesn't make much sense to me in this context...
From now on I would like you guys to address me as: "Dear Leader, who is a perfect incarnation of the appearance that a leader should have". Thanks in advance!
Q: There were 32 students in a hostel. Due to the admission of 7 new students the expenses of the mess were increased by $42 per day while the average expenditure per head diminished by $1. What was the original expenditure of the mess?
Now you only need to write down one more fact i.e. you need to make an equation containing E and the information of the minus one dollar per day on average.
Average here means price per person which is the expense divided by the number of people.
@FreakEnum I think I get something different but I might have got the sums wrong. Can you tell me the equation containing the one dollar difference information?
(Does really no one fancy a very short uniform convergence question? T_T)