All Questions

The following is inspired from the most recent riddle of the week of the German news magazine Der Spiegel. For any positive integer $n\in\mathbb{N}$, let $[n]$ denote the set $\{1,\ldots,n\}$. Let $...

Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$ $$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$ $$C_k = \frac{\sum_{i=1}^k(...

Let $$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$ $$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$ $$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...

Given $x \in (0,\frac{1}{2})$ and $y \in (0,\frac{1}{2}]$, what is the value of the following limit: $\lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}|?$ When $...

Let $X$ denote an $m\times n$ matrix and suppose that each value $x_{ij}$ is an integer that is selected uniformly at random from ${1,\dots,n}$, independently of all other values. If we fix $m$ and ...