If $C_r= ^{100}{C_r}$ , then $1.C^2_0 - 2.C^2_1 + 3.C^2_3 - 4.C^2_0 + 5.C^2_4 - .... + 101.C^2_{100}$ is equal to
${100.^{100}}{C_{50}}\,\,\,$
${51.^{100}}{C_{50}}\,\,\,$
${100.^{200}}{C_{100}}\,\,\,$
${51.^{200}}{C_{100}}\,\,\,$
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
In the polynomial $(x - 1)(x - 2)(x - 3).............(x - 100),$ the coefficient of ${x^{99}}$ is
Let n and k be positive integers such that $n \ge \frac{{k(k + 1)}}{2}$. The number of solutions $({x_1},{x_2},....{x_k})$, ${x_1} \ge 1,{x_2} \ge 2,....{x_k} \ge k,$ all integers, satisfying ${x_1} + {x_2} + .... + {x_k} = n$, is
Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta } $ for every value of $\theta $, then