Binomial Theorem - Algebra - Mathematics Previous Year Questions

Which of the following is the correct principle of Mathematical induction?

The coefficient of the term independent of \(x\) in the expansion \(\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}\) is

If the 2nd, 3rd and 4th terms in the expansion of \((a+b)^n\) be \(240,720\) and 1080 respectively, then the value of \((n, b, a)\) is

Last three digits in \((9)^{50}\) be

If \(x^n=a_0+a_1(1+x)+a_2(1+x)^2+\ldots \ldots \ldots+ a_n(1+x)^n=b_0+b_1(1-x)+b_2(1-x)^2+\ldots . .+ b_n(1-x)^n\), then for \(n=201,\left(a_{101}, b_{101}\right)\) is equal to

If the coefficients of $x^7$ and $x^8$ in the expansion of $\left[2+\frac{x}{3}\right]^n$ are equal, then value of $n$ is

If the coefficient of $x^2$ and $x^3$ in the expansion of $\left(1+8 x+b x^2\right)(1-3 x)^9$ in the power of $x$ are equal, then $b$ is

Coefficient of $x^3$ in the expansion of $\left(x^2-x+1\right)^{10}\left(x^2+1\right)^{15}$ is equal to