Question

The coefficient of $x^{3}$ in the expansion of $(1 - \frac{3}{4}x)^{\frac{1}{2}}$ is

• A. $\frac{27}{1024}$
• B. $\frac{-27}{1024}$
• C. $\frac{81}{1024}$
• D. $\frac{-81}{1024}$

Hint:

For $(1-x)^{n}$ with $n=1/2$, the terms alternate signs. The third power ($x^{3}$) will have a negative coefficient.

Answer 1

The Correct Option is B

Step 1: Concept
The binomial expansion for $(1 + y)^{n}$ when $n$ is a fraction is given by $1 + ny + \frac{n(n-1)}{2!}y^{2} + \frac{n(n-1)(n-2)}{3!}y^{3} + \dots$

Step 2: Meaning
In this problem, $n = \frac{1}{2}$ and $y = -\frac{3}{4}x$. We need to find the term containing $y^{3}$ to extract the coefficient of $x^{3}$.

Step 3: Analysis
The $x^{3}$ term is $\frac{(\frac{1}{2})(\frac{1}{2}-1)(\frac{1}{2}-2)}{3!} (-\frac{3}{4}x)^{3}$. Calculating the numerator: $(\frac{1}{2})(-\frac{1}{2})(-\frac{3}{2}) = \frac{3}{8}$. Dividing by $3!$ (which is 6): $\frac{3/8}{6} = \frac{3}{48} = \frac{1}{16}$. Calculating the power of $y$: $(-\frac{3}{4}x)^{3} = -\frac{27}{64}x^{3}$. Multiplying them: $\frac{1}{16} \times (-\frac{27}{64})x^{3} = -\frac{27}{1024}x^{3}$.

Step 4: Conclusion
The coefficient of $x^{3}$ is $-\frac{27}{1024}$. Final Answer: (B)