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 $S_n =$$\sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}}} $ and $T_n =$$\sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}}} $ then $\frac{{{T_n}}}{{{S_n}}}$ is equal to
Coefficient of $x^{64}$ in the expansion of $(x - 1)^2(x - 2)^3(x - 3)^4(x - 4)^5 .... (x - 10)^{11}$
The sum of all the coefficients in the binomial expansion of ${({x^2} + x - 3)^{319}}$ is
If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) ...... (1 + x + x^2 + x^3 + ...... + x^n)$
$\equiv a_0 + a_1x + a_2x^2 + a_3x^3 + ...... + a_mx^m$ then $\sum\limits_{r\, = \,0}^m {\,\,{a_r}}$ has the value equal to
The coefficient of $x^9$ in the polynomial given by $\sum\limits_{r - 1}^{11} {(x + r)\,(x + r + 1)\,(x + r + 2)...\,(x + r + 9)}$ is