Question 24
Consider the set $C=\{(x,y):x,y\text{ are integers and }0\le x\le y\le10\}$. An element $\mathbf u$ in $C$ is called beautiful if $\mathbf u=(u,u)$, where $u\in\{0,1,\ldots,10\}$. An element $\mathbf v$ in $C$ is called ugly if $\mathbf v=(v_1,v_2)$, where $0\le v_1\lt v_2\le10$ and $v_1,v_2\in\mathbb Z$. Two elements are drawn at random, one after another and without replacement, from the set $C$. Let $p$ be the probability that exactly one beautiful element and exactly one ugly element is drawn, such that both the beautiful and ugly element have exactly one number in common. An example of such an outcome is $(1,1),(1,3)$. Find the value of $\left\lceil\frac1p\right\rceil$.