Question 35
Let $X=\{1,2,3,4,5,6,7\}$, and let $\mathcal A=\{F_1,F_2,\ldots,F_n\}$ be a collection of distinct subsets of $X$ such that the intersection $F_i\cap F_j$ contains exactly one element whenever $i\ne j$. For each $i\in X$, let $r_i$ be the number of elements in $\mathcal A$ which contains $i$. Suppose $r_1=r_2=1$, $r_3=r_4=r_5=r_6=2$ and $r_7=4$. Find the value of $n^2-n$.