Let [t] denote the greatest integer t.

Question

Let $[t]$ denote the greatest integer $\le t$. If the constant term in the expansion of ${(3{x}^{2}-\frac{1}{2{x}^{5}})}^{7}$ is $\alpha $, then $[\alpha ]$ is equal to \_\_\_\_\_\_\_\_\_\_\_.

Answer: 1275

Accepted Answer

Let ${T}_{r+1}$ be the constant term.

${T}_{r+1}={}^{7}{C}_{r}{(3{x}^{2})}^{7-r}{(\frac{-1}{2{x}^{5}})}^{r}$

For constant term, power of $x$ should be zero.

$\begin{gathered} 14-2r-5r=0 \ \Rightarrow 14=7r\Rightarrow r=2 \end{gathered}$

Now, constant term $=\alpha $

$\begin{gathered} \Rightarrow {}^{7}{C}_{2}(3{)}^{5}{(\frac{-1}{2})}^{2}=\alpha \ \Rightarrow 21	imes 243	imes \frac{1}{4}=\alpha \ \Rightarrow [\alpha ]=[1275.75]=1275 \end{gathered}$

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