Question

If \( ^nC_{r-1}=36, ^nC_r=84, ^nC_{r+1}=126 \), then \( n= \)

• A. \( 3 \)
• B. \( 4 \)
• C. \( 8 \)
• D. \( 9 \)
• E. \( 10 \)

Hint:

Use ratios to eliminate factorial complexity.

Answer 1

The Correct Option is D

Concept: Ratio of consecutive binomial coefficients.

Step 1: Use: \[ \frac{^nC_r}{^nC_{r-1}}=\frac{n-r+1}{r} \]

Step 2: Substitute: \[ \frac{84}{36}=\frac{n-r+1}{r} \]

Step 3: Simplify: \[ \frac{7}{3}=\frac{n-r+1}{r} \Rightarrow 7r=3n-3r+3 \Rightarrow 10r=3n+3 \]

Step 4: Similarly: \[ \frac{126}{84}=\frac{n-r}{r+1} \Rightarrow \frac{3}{2}=\frac{n-r}{r+1} \]

Step 5: Solve: \[ 3(r+1)=2(n-r) \Rightarrow 3r+3=2n-2r \Rightarrow 5r=2n-3 \] Solve equations: \[ n=9 \]