In the expansion of (x(4/5) + x- (1/5))n , the coeficient of the 8t h and 19t h terms are equal. The term independent of x is given by

Question

In the expansion of (x(4/5) + x- (1/5))n , the coeficient of the 8t h and 19t h terms are equal. The term independent of x is given by (A)  27C21 (B)  25C20 (C)  25C21 (D)  27C22

Tardigrade Admin · Accepted Answer

Since there is no constant term the coefficient of

8^{th}

and

1 9^{th}

term are same as the binomial coefficients of

8^{th}

and

1 9^{th}

term.

_^{n} C_{7} =^{n} C_{18} \Rightarrow n = 7 + 18 = 25

T_{n + 1} =^{25} C_{r} (x^{\frac{4}{5}})^{25 - r} (r^{- \frac{1}{5}})^{r}

_^{25} C_{r} x^{\frac{4}{5} (25 - r) - \frac{r}{5}}

To be independent of

x, \frac{100 - 5 r}{5} = 0 \Rightarrow r = 20

Hence, the required term is

_^{25} C_{20}

Q. In the expansion of $(x^{\frac{4}{5}} + x^{- \frac{1}{5}})^{n}$ , the coeficient of the $8^{t h}$ and $1 9^{t h}$ terms are equal. The term independent of $x$ is given by

$_^{27} C_{21}$

$_^{25} C_{20}$

$_^{25} C_{21}$

$_^{27} C_{22}$

Q. In the expansion of (x54​+x−51​)n , the coeficient of the 8th and 19th terms are equal. The term independent of x is given by

_27C21​

_25C20​

_25C21​

_27C22​

Q. In the expansion of $(x^{\frac{4}{5}} + x^{- \frac{1}{5}})^{n}$ , the coeficient of the $8^{t h}$ and $1 9^{t h}$ terms are equal. The term independent of $x$ is given by

$_^{27} C_{21}$

$_^{25} C_{20}$

$_^{25} C_{21}$

$_^{27} C_{22}$