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  4. If the constant term in the binomial expansion of (x2-(1/x))n, n ∈ N is 15 then the value of n is equal to

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Q. If the constant term in the binomial expansion of $\left(x^2-\frac{1}{x}\right)^n, n \in N$ is 15 then the value of $n$ is equal to

Binomial Theorem

Solution:

$T_{r+1}$ in $\left(x^2-\frac{1}{x}\right)^n$ is
${ }^n C_r\left(x^2\right)^{n-r}(-1)^r x^{-r} $
$={ }^n C_r x^{2 n-3 r}(-1)^r$
Constant term $={ }^n C_r(-1)^r$ if $2 n=3 r$ i.e. coefficient of $x=0$ hence ${ }^n C_{2 n / 3}(-1)^{2 n / 3}=15={ }^6 C_4 \Rightarrow n=6$