Discussion
The coefficient of x3 in \(\left(\sqrt{x^{5}}+\frac{3}{\sqrt{x^{3}}}\right)^{6}\) is
Answer: D
The coefficient of x3 is 540.
The general term of \((x^{5/2} + 3x^{-3/2})^{6}\) is:
\[ \begin{aligned} T_{r+1} &= \binom{6}{r}(x^{5/2})^{6-r}(3x^{-3/2})^{r} \\ &= \binom{6}{r}3^{r}x^{15 - 4r} \end{aligned} \]
For x3, 15 - 4r = 3, so r = 3, and the coefficient is \(\binom{6}{3} \times 3^{3} = 20 \times 27 = 540\). 120 forgets the 33 (or uses a wrong binomial), and 0 would mean no term has power 3, which is not the case.
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