Question

In the expansion of \(\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x>0\), if the term independent of \(x\) is \((221)k\), then \(k\) is equal to:

• A. 84
• B. 78
• C. 168
• D. 198

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the term independent of \(x\) in a binomial expansion, write the general term \(T_{r+1}\) and set the exponent of \(x\) to zero to find the value of \(r\).

Step 2: Key Formula or Approach:
1. \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\).
2. Solve \(n \cdot p - r(p + q) = 0\) for the term independent of \(x\) in \((x^p + x^{-q})^n\).

Step 3: Detailed Explanation:
Here \(n=18, p=1, q=1/2\).
\(r = \frac{18 \times 1}{1 + 1/2} = \frac{18}{1.5} = 12\).
The term is \(T_{13} = \binom{18}{12} (9x)^{18-12} (-\frac{1}{3\sqrt{x}})^{12}\).
\(T_{13} = \binom{18}{12} \cdot 9^6 \cdot x^6 \cdot (\frac{1}{3})^{12} \cdot x^{-6} = \binom{18}{6} \cdot (3^2)^6 \cdot 3^{-12} = \binom{18}{6}\).
\(\binom{18}{6} = \frac{18 \cdot 17 \cdot 16 \cdot 15 \cdot 14 \cdot 13}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 18564\).
Given \(221k = 18564 \implies k = \frac{18564}{221} = 84\).

Step 4: Final Answer:
The value of \(k\) is 84.