If the number of integral terms in the expansion

Question:

If the number of integral terms in the expansion of $\left(3^{1 / 2}+5^{1 / 8}\right)^{n}$ is exactly 33 , then the least value of $n$ is :

  1. 264

  2. 256

  3. 128

  4. 248


Correct Option: , 2

Solution:

$\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}(3)^{\frac{\mathrm{n}-\mathrm{r}}{2}}(5)^{\frac{\mathrm{r}}{8}} \quad(\mathrm{n} \geq \mathrm{r})$

Clearly $\mathrm{r}$ should be a multiple of $8 .$

$\because$ there are exactly 33 integral terms

Possible values of $\mathrm{r}$ can be

$0,8,16, \ldots \ldots \ldots, 32 \times 8$

$\therefore$ least value of $\mathrm{n}=256$

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