The number of integral terms in the expansion of $\left(3^{1 / 2}+2^{1 / 2}\right)^{500}$ is

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Professor PC Thomas Classes · Answer

$ \begin{aligned} & \mathrm{T}_{\mathrm{r}+1} 	ext { in }(\sqrt{3}+\sqrt{2})^{500} \ & ={ }^{500} \mathrm{C}_{\mathrm{r}}(\sqrt{3})^{500-\mathrm{r}}(\sqrt{2})^{\mathrm{x}} \ & ={ }^{500} \mathrm{C}_{\mathrm{r}} 3^{\frac{500-\mathrm{r}}{2}} \cdot 2^{\frac{\mathrm{r}}{2}} \end{aligned} $ \  $	herefore \mathrm{r}$ is even $	herefore r$ can take values \ $ 0,2,4, \ldots 498,500 . $ ∴ There are 251 integral terms.