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 1h comment What is the value of $k^2$ You can use \left and \right to make the parentheses adapt to their content. 3h comment Taylor expand $\ln(x) - \ln(1-y)$ with respect to $\ln(x')$ and $\ln(y')$ (Part of what's confusing is that the partial derivatives seem to imply a two-dimensional coordinate transformation, but then you write that you expand around "the points $\ln(x')$ and $\ln(y')$" in the plural, and not "the point $(\ln(x'),\ln(y')$"). 3h comment Taylor expand $\ln(x) - \ln(1-y)$ with respect to $\ln(x')$ and $\ln(y')$ It's not clear to me whether you're considering $x'$ and $y'$ as specific values of $x$ and $y$ or as a pair of new variables. The notation with primes seems to favour the latter interpretation, but the only way I can make sense of this is to consistently apply the former. Why does the constant term have $\ln(y')$ and not $-\ln(1-y')$? Perhaps it would help if you explicate how you expanded $f(\ln(x),\ln(y))$. Principally speaking, you can certainly treat the logarithms as the independent variables and expand in them, but as far as I can tell you haven't consistently done so. 9h comment What is probability that out of the first half on N objects, none will be matched with their own label? Thanks; I wasn't aware of this connection between Möbius inversion and inclusion-exclusion. 9h comment What is probability that out of the first half on N objects, none will be matched with their own label? @MarkoRiedel: The extra factor $\frac1{N!}$ is there because the question asked for the probability and you gave the count instead; other than that the results are identical. 9h comment What is probability that out of the first half on N objects, none will be matched with their own label? You can use \ldots instead of ... to get the right spacing. 11h comment You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life. @Ant: In the numerator, it's $((63-0)(63-18))((63-1)(63-17))\cdots((63-8)(63-10))(63-9)\approx(63-9)^{19}$; in the denominator it's $((1\cdot19)(2\cdot18)\cdots(9\cdot11)10\approx10^{19}$. 14h comment Painting the plane red and blue: Is it possible for each unit circumference to contain exactly $n$ blue points? Perhaps you could share your proof for $n=1$? Since this is the one you want to generalise. 15h comment Painting the plane red and blue: Is it possible for each unit circumference to contain exactly $n$ blue points? I think $8$ upvotes within $17$ minutes resolves the "improper use of Stackexchange" issue :-) 16h comment Order Statistics with two Groups of Draws @testrado: Could you give specific values of $m,n,a,x$ for which you get probabilities $\gt1$? I couldn't find any in a brief check. I believe the factor $m$ in that first fraction is right; it takes into account that you can select any of the $m$ samples not to be among the $k$ values. 18h comment Number of non periodic strings Hi -- welcome to math.SE! This is not string theory. Please take note of the tag summaries when choosing tags. 19h comment Prove that the core of this game is empty Hi -- welcome to math.SE! Here's a reference and tutorial for typesetting math on this site. 19h comment A sum of Stirling numbers of the second kind This could be a start: \begin{align} S(N) &= \sum_{n=N}^\infty \sum_{k=N}^n \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} (1+j)^n \frac{t^n}{n!}\\ &=\sum_{n=N}^\infty \sum_{k=N}^n \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} \sum_{l=0}^n\binom nlj^l \frac{t^n}{n!}\\ &=\sum_{n=N}^\infty \sum_{k=N}^n \sum_{l=0}^n\binom nl\frac{t^n}{n!}\sum_{j=0}^k \binom{k}{j} (-1)^{k-j}j^l \\ &=\sum_{n=N}^\infty \sum_{k=N}^n \sum_{l=0}^n\binom nl\frac{t^n}{n!}k!\left\{l\atop k\right\}\\ \end{align} 20h comment Separating Heavier from the Lighter Balls Also, you ask about the minimum number of weighings, but your further considerations show that you're actually interested in minimising the expected number of weighings. 20h comment Separating Heavier from the Lighter Balls Where it says "$30\%$" I think you mean $\frac13$. (By the way this is a very well-posed question.) 21h comment As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? (The expectation of the determinant is shown to be $0$, so the variance is the expectation of its square, which is also calculated.) 23h comment As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? The question as clarified in your comment is answered in that duplicate question. 1d comment Determine probability of getting a desired outcome You also need to specify how the list you're presented comes about. Is it presented by an adversary, or drawn randomly? If randomly, with which distribution? Can the same option appear more than once in the list? 1d comment Probability problem with combination of poisson and binomial distributions You can use \left and \right to let the parentheses adjust to their content. 1d comment Latin squares using fixed word lists No -- if you can prove that answer, it would be best to write it up as an answer and accept it.