Discussion
For a BIBD with parameters \((v, b, r, k, \lambda)\) to exist, the condition \(b \ge v\) is :
Answer: C
Fisher's inequality \(b \ge v\) is necessary but not sufficient.
Every BIBD satisfies \(b \ge v\) (Fisher's inequality), so a parameter set with \(b \lt v\) cannot exist: the condition is necessary.
It is not sufficient. The parameters \(v = 15\), \(b = 21\), \(r = 7\), \(k = 5\), \(\lambda = 2\) satisfy \(vr = bk\), \(\lambda(v - 1) = r(k - 1)\) and \(b \ge v\), yet no such design exists. Similarly, by the Bruck-Ryser-Chowla theorem there is no symmetric (22, 7, 2) design.
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