• Balanced Incomplete Block Design (BIBD) Verification:
• BIBD is defined by 5 parameters: $v$ treatments, $b$ blocks, $k$ plots per block ($k \lt v$), $r$ replications per treatment, and $\lambda$ pairs per block, satisfying:

• Evaluating Treatment Frequencies in Existing Blocks:
• Replications ($r = 3$ each):
• Treatment 1 appears in: Block 1, Block 3 (current count = 2 $\implies$ needed in Block 4)
• Treatment 2 appears in: Block 1, Block 2 (current count = 2 $\implies$ needed in Block 4)
• Treatment 3 appears in: Block 2, Block 3 (current count = 2 $\implies$ needed in Block 4)
• Treatment 4 appears in: Block 1, Block 2, Block 3 (current count = 3 $\implies$ fully replicated, cannot appear in Block 4)
• Pair Concurrence ($\lambda = 2$):
• Pair (1, 4): Blocks 1, 3 (count = 2)
• Pair (2, 4): Blocks 1, 2 (count = 2)
• Pair (3, 4): Blocks 2, 3 (count = 2)
• Pair (1, 2): Block 1 only (count = 1 $\implies$ must co-occur in Block 4)
• Pair (2, 3): Block 2 only (count = 1 $\implies$ must co-occur in Block 4)
• Pair (1, 3): Block 3 only (count = 1 $\implies$ must co-occur in Block 4)
• Therefore, Block 4 must contain treatments {1, 2, 3} (Option 3: 2 3 1).

Correct option: (3) 231

• Associacy Matrices in Two-Associate Class PBIBD:
• In a two-associate PBIBD, parameters satisfy the identity relations:

• Given $n_1 = 6$ and $n_2 = 2$:
• For $P^1$ (first associates):
• Row 1: $p_{11}^1 + p_{12}^1 = 6 - 1 = 5 \implies 3 + 2 = 5$
• Row 2: $p_{21}^1 + p_{22}^1 = n_2 = 2 \implies 2 + 0 = 2$
• Hence, $P^1 = \begin{pmatrix} 3 & 2 \\ 2 & 0 \end{pmatrix}$
• For $P^2$ (second associates):
• Row 1: $p_{11}^2 + p_{12}^2 = n_1 = 6 \implies 6 + 0 = 6$
• Row 2: $p_{21}^2 + p_{22}^2 = n_2 - 1 = 2 - 1 = 1 \implies 0 + 1 = 1$
• Hence, $P^2 = \begin{pmatrix} 6 & 0 \\ 0 & 1 \end{pmatrix}$
• Conclusion: Matches Option 2.

Correct option: (2) ana p?= (5 ° 21 ol 2 as 4 ana p?= (¢ e 20 o1