If $C_{x} \equiv^{25} C_{x}$ and $\mathrm{C}_{0}+5 \cdot \mathrm{C}_{1}+9 \cdot \mathrm{C}_{2}+\ldots .+(101) \cdot \mathrm{C}_{25}=2^{25} \cdot \mathrm{k}$ then $\mathrm{k}$ is equal to
$42$
$45$
$51$
$48$
If $n$ is an integer greater than $1$, then $a{ - ^n}{C_1}(a - 1){ + ^n}{C_2}(a - 2) + .... + {( - 1)^n}(a - n) = $
The value $\sum \limits_{ r =0}^{22}{ }^{22} C _{ r }{ }^{23} C _{ r }$ is $.......$
${C_0} - {C_1} + {C_2} - {C_3} + ..... + {( - 1)^n}{C_n}$ is equal to
If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$ equals
The expression $x^3 - 3x^2 - 9x + c$ can be written in the form $(x - a)^2 (x - b)$ if the values of $c$ is