Question

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

• A. \( 9 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)
• E. \( 6 \)

Hint:

Successive binomial coefficients in the middle of a row of Pascal's triangle always show specific ratios. Working with these ratios is much faster than expanding the full factorial definitions.

Answer 1

The Correct Option is B

Concept: The relationship between consecutive binomial coefficients \( \binom{n}{r} \) is governed by the ratio formula: \( \frac{\binom{n}{r}}{\binom{n}{r-1}} = \frac{n-r+1}{r} \). By creating a system of equations from the given values, we can solve for both \( n \) and \( r \).

Step 1: Set up ratios from the given values.
Using the first two values: \[ \frac{nC_r}{nC_{r-1}} = \frac{84}{36} = \frac{7}{3} \] \[ \frac{n-r+1}{r} = \frac{7}{3} \implies 3n - 3r + 3 = 7r \implies 3n - 10r = -3 \quad \cdots (1) \]

Step 2: Set up a second ratio.
Using the next two values: \[ \frac{nC_{r+1}}{nC_r} = \frac{126}{84} = \frac{3}{2} \] \[ \frac{n-(r+1)+1}{r+1} = \frac{3}{2} \implies \frac{n-r}{r+1} = \frac{3}{2} \] \[ 2n - 2r = 3r + 3 \implies 2n - 5r = 3 \quad \cdots (2) \]

Step 3: Solve the system for \( r \).
Multiply equation (2) by 2: \[ 4n - 10r = 6 \] Subtract equation (1) \( (3n - 10r = -3) \) from this new equation: \[ (4n - 3n) + (-10r - (-10r)) = 6 - (-3) \implies n = 9 \] Substitute \( n = 9 \) into equation (2): \[ 2(9) - 5r = 3 \implies 18 - 5r = 3 \implies 5r = 15 \implies r = 3 \]