Binomial Theorem for any Index

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If the expansion in powers of $x$ of the function $\frac{1}{( 1 - a x ) ( 1 - b x )}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} ....$ then $a_{n}$ is

\frac{b ^{n} - a ^{n}}{b - a}

\frac{a ^{n} - b ^{n}}{b - a}

\frac{a ^{n + 1} - b ^{n + 1}}{a - b}

\frac{b ^{n + 1} - a ^{n + 1}}{b - a}

If $x$ is so small that $x^{3}$ and higher powers of $x$ may be neglected, then $\frac{( 1 + x ) ^{3/2} - ( 1 + \frac{1}{2} x ) ^{3}}{( 1 - x ) ^{1/2}}$ may be approximated as

1 - \frac{3}{8} x^{2}

3 x + \frac{3}{8} x^{2}

- \frac{3}{8} x^{2}

\frac{x}{2} - \frac{3}{8} x^{2}

If $x$ is positive, the first negative term in the expansion of $(1 + x)^{27/5}$ is

JEE Main 2019 (9 January)

The coefficient of $t^{4}$ in the expansion of $(\frac{1 - t ^{6}}{1 - t})^{3}$ is

JEE Main 2020 - 5 Sep (Evening)

The coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2} + x^{3})^{6}$ in powers of $x$ , is