${C_0}{C_r} + {C_1}{C_{r + 1}} + {C_2}{C_{r + 2}} + .... + {C_{n - r}}{C_n}$=
$\frac{{(2n)!}}{{(n - r)\,!\,(n + r)!}}$
$\frac{{n!}}{{( - r)!(n + r)!}}$
$\frac{{n!}}{{(n - r)!}}$
None of these
If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is .... .
If $^{20}{C_1} + \left( {{2^2}} \right){\,^{20}}{C_3} + \left( {{3^2}} \right){\,^{20}}{C_3} + \left( {{2^2}} \right) + ..... + \left( {{{20}^2}} \right){\,^{20}}{C_{20}} = A\left( {{2^\beta }} \right)$, then the ordered pair $(A, \beta )$ is equal to
What is the sum of the coefficients of ${({x^2} - x - 1)^{99}}$
Suppose $\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}$. Then the value of $\alpha$ is $............$
If $(1 + x - 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + .........$ then $a_0 - a_1 + a_2 - a_3 + ..... $ ends with