Question

If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), and \( ^nC_{r+1} = 126 \), then the value of \( nr^2 \) is:

• A. \( 243 \)
• B. \( 9 \)
• C. \( 27 \)
• D. \( 81 \)

Hint:

Binomial Ratio Equations}
Use ratio identities of binomial coefficients to build equations.
Solve system of equations for \( n \) and \( r \).
Common trick: use known ratios to form simultaneous linear equations.

Answer 1

The Correct Option is D

Given: \[ ^nC_{r-1} = 36, \quad ^nC_r = 84, \quad ^nC_{r+1} = 126. \] Use the property: \[ \frac{^nC_r}{^nC_{r-1}} = \frac{n - r + 1}{r} \Rightarrow \frac{84}{36} = \frac{n - r + 1}{r} \Rightarrow \frac{7}{3} = \frac{n - r + 1}{r} \Rightarrow 7r = 3n - 3r + 3 \Rightarrow 10r = 3n + 3 \quad \cdots (1) \] Also: \[ \frac{^nC_{r+1}}{^nC_r} = \frac{n - r}{r + 1} \Rightarrow \frac{126}{84} = \frac{n - r}{r + 1} \Rightarrow \frac{3}{2} = \frac{n - r}{r + 1} \Rightarrow 3r + 3 = 2n - 2r \Rightarrow 5r = 2n - 3 \quad \cdots (2) \] Solving equations (1) and (2) simultaneously gives: \[ r = 3, \quad n = 9. \] Hence: \[ nr^2 = 9 \times 9 = 81. \]