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23h
reviewed Leave Open What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
23h
comment How to deduce the formula for arrangement in groups?
@user36790: If we are thinking of dividing the people into $r$ named groups, we are essentially counting onto functions from an $n$-element set to an $r$-element set. This brings up the Stirling Numbers of the Second Kind (please see Wikipedia).
23h
comment How to deduce the formula for arrangement in groups?
@Marc: In addition to counting order within groups, one also needs to assume that the groups are named, making $ab|c$ is different from $c|ab$.
23h
comment How to deduce the formula for arrangement in groups?
Note that one needs to make a peculiar interpretation of the wording to get the formula. For example, there are $3$ ways to divide the people $a,b,c$ into $2$ non-empty groups. But the formula gives $12$.
1d
revised Four different green balls and red balls
deleted 18 characters in body
1d
answered Four different green balls and red balls
1d
comment What is the probability of a randomly chosen bit string of length 8 does not contain 2 consecutive 0's?
I assume $2$ consecutive $0$'s means at least $2$. You are right, it is easier to count the complement. If $a_n$ is the number of bit strings of length $n$ where no two consecutive $0$'s occur, show that $a_n=a_{n-1}+a_{n-2}$. The $a_n$ will turn out to be Fibonacci numbers. (It is a good idea as a preliminary step to find $a_0,a_1,a_2,a_3$ and maybe a few others by explicit listing.)
1d
reviewed Leave Open How calculate the probability that there is a row in which there is no silver coin?
1d
comment Recurrence relation for a mortgage
Either compute the $a_k$, using the recurrence, until $a_k \lt \frac{2000}{1.01}$, ( nuisance, since the mortgage continues for a long time) and then it will be easy, or find a general formula for $a_k$ (forgetting about the fact the mortgage ends). If you find such a formula, you will be able to find the largest $k$ such that $a_k\lt \frac{2000}{101}$ by using logarithms.
1d
comment How calculate the probability that there is a row in which there is no silver coin?
@Yann: I am using "combinations" while your approach uses "permutations." That is another way of doing it. But then the denominator changes to $(n^2)(n^2-1)\cdots (n^2-n+1)$ and we get the same answer.
1d
comment How can I find the radius and interval of convergece of $\sum_{n=0}^\infty {(x+5)^n} $, and for what value of x does the series converge?
Geometric series.
1d
comment How calculate the probability that there is a row in which there is no silver coin?
Forget about the rest of the coins, let us distribute the $n$ silvers first. We find the probability of the complement, the event that all the silvers are in different rows. There are $\binom{n^2}{n}$ equally likely ways to choose the locations of the $n$ silvers. And there are $n^n$ ways to choose $1$ location in each row.
1d
comment Recurrence relation for a mortgage
Close. Until the last payment, which will be smaller, we have $a_n=(1.1)a_{n-1}-2000$. Similar to yours, but you had a minus sign.
1d
answered A question on the requirement of a quadrilateral being an adventitious quadrangle
1d
answered number of weak compositions modulo prime $p$
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comment Drawing 6 balls of different colours
Please see Wikipedia, Stars and Bars. The number of balls of each colour is irrelevant, as long as there are at least $6$ of each.
2d
comment Prove that there are infinitely many primes in $\mathbf Q[\sqrt{d}]$.
You are welcome. Once I found out that you (Stark) really meant irreducibles, I was hoping someone would flesh out the details of the proof. The theorem has to be stated precisely, particularly for those $d$ in which the ring has infinitely many units.
2d
comment Prove that there are infinitely many primes in $\mathbf Q[\sqrt{d}]$.
To prove that there are infinitely many irreducibles (what Stark calls primes), yes. Of course that is a lot simpler than the proof I wrote down, since there I was proving there are infinitely many primes in the modern sense of prime.
2d
comment Prove that there are infinitely many primes in $\mathbf Q[\sqrt{d}]$.
Infinitely many irreducibles. But as mixedmath discovered, in Stark's book the word prime is used to mean what everybody nowadays calls irreducible. Note I had a typo, it should be $N(\pi_1\pi_2\cdots \pi_n)+1$ (there was a parenthesis missing).
2d
awarded  Enlightened