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In a BIBD, let

$a$ denote the number of treatments;

$b$ denote the number of blocks;

$k$ denote the number of treatments exactly in each block;

$\lambda$ is the number of times each pair of treatments appears in the same block.

Each treatment occurs $r$ times.

How are the relations $r=\binom{a-1}{k-1}$ and $\lambda=\binom{a-2}{k-2}$ obtained?

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Can you give references for your claims? I don't think they are true, maybe they refer to some specific subset of BIBD's.

The number of experimental runs (total number of points) in A BIBD is $n$ which satisfies $n= ar = bk$ by simple combinatorics. We have Fisher's inequality https://en.wikipedia.org/wiki/Fisher's_inequality that $a \le b$, the number of treatments cannot exceed the number of blocks.

Less obvious is the relation $r (k-1) = \lambda (a-1)$, which can be proved by counting concentrating only on the blocks which contain one given treatment.

You give $r = \binom{a-1}{k-1}$ which I can prove if I assume the blocks consists of all $k$-subsets from the $a$ treatments, but many BIBD's are smaller than that.

One example which give counterexamples is $$ 0000011122 \\ 1123423433 \\ 2345554545 $$ (each column is a block) which has $$ a=6 \\ k=3 \\ b=10 \\ r=5 \\ \lambda=2 $$ and you can verify that $$ \binom{a-1}{k-1}=\binom{5}{2}=10 \not = 5 = r \\ \binom{a-2}{k-2}=\binom{4}{1}=4 \not= 2 = \lambda $$

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