Balanced incomplete block design 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?
 A: 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
$$
\binom{a-1}{k-1}=\binom{5}{2}=10 \not = 5 = r \\
\binom{a-2}{k-2}=\binom{4}{1}=4 \not= 2 = \lambda
$$
