Question

If the coefficients of the \( (2r + 6)^{th \) and \( (r - 1)^{th} \) terms in the expansion of \( (1 + x)^{21} \) are equal, then the value of \( r \) is:}

• A. \( 7 \)
• B. \( 5 \)
• C. \( 6 \)
• D. \( 8 \)

Hint:

Use the symmetric property of binomial coefficients \( \binom{n}{k} = \binom{n}{n-k} \) to simplify expressions and solve for unknowns.

Answer 1

The Correct Option is C

Step 1: Understanding binomial coefficients
The general term in the expansion of \( (1 + x)^n \) is given by: \[ T_k = \binom{n}{k} x^k. \] The coefficients of the \( (2r+6)^{th} \) and \( (r-1)^{th} \) terms are: \[ \binom{21}{2r+6}, \quad \binom{21}{r-1}. \] Since these coefficients are equal, we set up the equation: \[ \binom{21}{2r+6} = \binom{21}{r-1}. \] Step 2: Using binomial coefficient property
Using the symmetric property of binomial coefficients: \[ \binom{n}{k} = \binom{n}{n-k}, \] we get: \[ \binom{21}{2r+6} = \binom{21}{21-(2r+6)} = \binom{21}{15-2r}. \] Since \( \binom{21}{2r+6} = \binom{21}{r-1} \), we equate: \[ r - 1 = 15 - 2r. \] Step 3: Solving for \( r \)
Rearrange the equation: \[ r + 2r = 15 + 1. \] \[ 3r = 16. \] \[ r = 6. \] Step 4: Conclusion
Thus, the final answer is: \[ \boxed{6}. \]