Combinatorics question

Problem 3. (10 marks) Let (V, B) be a 2 - (2n + 1, n, )) block design and let a @ V be a new variety. Define V := V U {a}, and B := B1 U B2 where Bi := {V \\ B : BE B } and B2 := { B U {a} : BE B}. Prove that (V, B) is a 3 - (2n + 2, n + 1, 1) block design. Hint. Your main task is to show that, given arbitrarily chosen T C V of size 3, T lies in precisely > blocks of (V, B). There are 2 cases: (1) a E T (easy) and (2) a # T (harder). For case (2) it suffices to show that #1 + (2 = A, where u is the number of blocks B E B, with T C B. You will find the Inclusion-Exclusion Principle useful for this case as also the values of A, for s = 1, 2 where As := ( k - s is the number of blocks in B (with v, k, t being the parameters of (V, B) given above) containing any given set S C V of size s