1. An $(l, a)$-design of a set is a collection of subsets of that set such that each subset contains exactly $l$ elements and that no two of the subsets share more than $a$ elements. How many $(2,1)$-designs are there of a set containing 8 elements?
Solution: There are $\binom{8}{2}=28$ 2-element subsets. Any two distinct such subsets have at most 1 common element; hence, for each subset, we can decide independently whether or not it belongs to the design, and we thus obtain $2^{28}$ designs.