If the constant term in the binomial expansion of $$x2$$−$$1xn$$,n∈N is $$15$$, then the value of n is equal $$to_____$$.

$$Tr+1=nCrx2n$$−$$r($$−$$1)rx$$−r $$=nCrx2n$$−$$3r($$−$$1)rConstant$$ term $$=nCr($$−$$1)r$$ if $$2n=3r$$ i.e., coefficient of $$x=0Hence$$, $$nC2n/3($$−$$1)2n/3=15=6C$$ ,$$n=6$$

Binomial theorem for positive integral Index

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Question

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

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Solution

$\begin{array}{l} T_{r + 1} =^{n} C_{r} {(x^{2})}^{n - r} (- 1)^{r} x^{- r} \\ =^{n} C_{r} x^{2 n - 3 r} (- 1)^{r} \end{array}$
Constant term $=^{n} C_{r} (- 1)^{r} if 2 n = 3 r i.e., coefficient of x = 0$
Hence, $^{n} C_{2 n / 3} (- 1)^{2 n / 3} = 15 =^{6} C, n = 6$

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