Question

If the coefficients of the \( r^{th} \), \( (r+1)^{th \), and \( (r+2)^{th} \) terms in the expansion of \( (1 + x)^n \) are in the ratio \( 4:15:42 \), then \( n - r \) is:}

• A. \( 18 \)
• B. \( 15 \)
• C. \( 14 \)
• D. \( 17 \)

Hint:

For binomial coefficient ratios, use the property \( \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{n-r}{r+1} \) to establish equations and solve for unknowns.

Answer 1

The Correct Option is C

Step 1: Understanding binomial coefficients
The general term in the binomial expansion of \( (1 + x)^n \) is given by: \[ T_k = \binom{n}{k} x^k. \] The coefficients of the \( r^{th} \), \( (r+1)^{th} \), and \( (r+2)^{th} \) terms are given by: \[ \binom{n}{r}, \quad \binom{n}{r+1}, \quad \binom{n}{r+2}. \] We are given the ratio: \[ \binom{n}{r} : \binom{n}{r+1} : \binom{n}{r+2} = 4:15:42. \] Step 2: Expressing binomial coefficients as ratios
Using the property of binomial coefficients: \[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{n-r}{r+1}, \] \[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{n-r-1}{r+2}. \] Thus, we obtain the equations: \[ \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{15}{4} = \frac{n-r}{r+1}, \] \[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{42}{15} = \frac{n-r-1}{r+2}. \] Step 3: Solving for \( n - r \)
From the first equation: \[ (n - r) = \frac{15}{4} (r + 1). \] From the second equation: \[ (n - r - 1) = \frac{42}{15} (r + 2). \] Solving these equations simultaneously gives: \[ n - r = 14. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{14}. \]