In the question, three series I, II and III are given. Find the value of $x$, $y$ and $z$ to establish the correct relation among them and choose the correct option. (i) 12, 24, 8, $x$, 6.4, 38.4 (ii) 11, 20, $y$, 50, 71, 96 (iii) 3, 4, 6, 12, $z$, 156

### Concept & Multi-Series Number Patterns Solve for unknown variables by determining the unique arithmetic, geometric, or difference-based rule governing each distinct series. ### Step-by-Step Solution * **Series I Pattern (Alternating Multiply/Divide):** $12, 24, 8, x, 6.4, 38.4$. * $12 \times 2 = 24$ * $24 \div 3 = 8$ * $8 \times 4 = 32 \implies x = 32$ * $32 \div 5 = 6.4$ (Pattern confirmed) * **Series II Pattern (Double Difference):** $11, 20, y, 50, 71, 96$. * First differences: $9, (y-20), (50-y), 21, 25$. * Notice the last two differences are $21$ and $25$ (a gap of 4). * Assume an arithmetic progression for differences: $9, 13, 17, 21, 25$. * $11 + 9 = 20$; $20 + 13 = 33 \implies y = 33$. * Check next: $33 + 17 = 50$. (Pattern confirmed) * **Series III Pattern (Factorial Differences):** $3, 4, 6, 12, z, 156$. * First differences: $1, 2, 6, (z-12), (156-z)$. * Notice $1=1!$, $2=2!$, $6=3!$. The next differences must be $4! = 24$ and $5! = 120$. * $12 + 24 = 36 \implies z = 36$. * Check next: $36 + 120 = 156$. (Pattern confirmed) * **Establishing Relation:** * $x = 32$, $y = 33$, $z = 36$. * Thus, $32 < 33 < 36$, which translates to $x < y < z$. ### Exam Strategy & Shortcut Look for highly recognizable numbers like factorials (1, 2, 6, 24) or clear geometric jumps to identify the pattern type quickly without full trial and error. ### Common Pitfall Misidentifying Series I as having addition/subtraction. The decimal values (6.4, 38.4) strongly hint at multiplication/division. ### Final Answer Therefore, the correct answer is **$x < y < z$**.