Find the coefficient of $a^{4}$ in the product $(1+2a)^{4}(2-a)^{5}$ using the binomial theorem.

Let the smallest value of $k \in N$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$, is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$, be $p$. Then the value of $p+n$ is:

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+.......+x^{1000}$ is

If ${T_0}, {T_1}, {T_2}, \dots, {T_n}$ represent the terms in the expansion of ${(x + a)^n}$,then $({T_0} - {T_2} + {T_4} - \dots)^2 + ({T_1} - {T_3} + {T_5} - \dots)^2 = $

The coefficient of $x^{13}$ in the expansion of $(1 - x)^5(1 + x + x^2 + x^3)^4$ is :-

The sum of the series $2 \times 1 \times {}^{20}C_4 - 3 \times 2 \times {}^{20}C_5 + 4 \times 3 \times {}^{20}C_6 - 5 \times 4 \times {}^{20}C_7 + \dots + 18 \times 17 \times {}^{20}C_{20}$ is equal to: