• First-Order Inclusion Probabilities ($\pi_k$):
• Population: $U = \{1, 2, 3\}$.
• Possible samples and their probabilities:

• Evaluating Inclusion Probabilities ($\pi_k = \sum_{s \ni k} P(s)$):
• $\pi_1 = P(\{1, 2\}) + P(\{1, 3\}) = 1/2 + 1/4 = 3/4$ (True)
• $\pi_2 = P(\{1, 2\}) + P(\{2, 3\}) = 1/2 + 1/4 = 3/4$
• $\pi_3 = P(\{1, 3\}) + P(\{2, 3\}) = 1/4 + 1/4 = 1/2$ (True)
• Fundamental Theorem of Inclusion Probabilities:
• For any sampling design with fixed sample size $n$, the sum of first-order inclusion probabilities equals the sample size:

• Here $n = 2$, so $\pi_1 + \pi_2 + \pi_3 = 3/4 + 3/4 + 1/2 = 2$.
• Statement in Option 3 asserts that $\pi_1 + \pi_2 + \pi_3 = 1$, which is mathematically NOT true.
• Conclusion: Option 3 is NOT true.

Correct option: (3) 1441 +1y=1

• PPS Selection via Cumulative Total Method:
• Number of trees in orchards 1 to 10: $150, 50, 80, 100, 200, 160, 40, 220, 60, 140$.
• Cumulative Totals and Selection Intervals:
• Orchard 1: $150 \implies [1 - 150]$
• Orchard 2: $150 + 50 = 200 \implies [151 - 200]$
• Orchard 3: $200 + 80 = 280 \implies [201 - 280]$
• Orchard 4: $280 + 100 = 380 \implies [281 - 380]$
• Orchard 5: $380 + 200 = 580 \implies [381 - 580]$
• Orchard 6: $580 + 160 = 740 \implies [581 - 740]$
• Orchard 7: $740 + 40 = 780 \implies [741 - 780]$
• Orchard 8: $780 + 220 = 1000 \implies [781 - 1000]$
• Orchard 9: $1000 + 60 = 1060 \implies [1001 - 1060]$
• Orchard 10: $1060 + 140 = 1200 \implies [1061 - 1200]$
• Matching Chosen Random Numbers:
• Draw 1 (600): Falls in $[581 - 740] \implies$ Orchard 6
• Draw 2 (300): Falls in $[281 - 380] \implies$ Orchard 4
• Draw 3 (650): Falls in $[581 - 740] \implies$ Orchard 6
• Draw 4 (850): Falls in $[781 - 1000] \implies$ Orchard 8
• Conclusion: The orchards selected are 6, 4, 6, 8 (Option 2).

Correct option: (2) then the orchards selected are 6, 4, 6, 8