The value of $^{4n}{C_0}{ + ^{4n}}{C_4}{ + ^{4n}}{C_8} + ....{ + ^{4n}}{C_{4n}}$ is
${2^{4n - 2}} + {( - 1)^n}{2^{2n - 1}}$
${2^{4n - 2}} + {2^{2n - 1}}$
${2^{2n - 1}} + {( - 1)^n}\,{2^{4n - 2}}$
None of these
The sum of the coefficients in the expansion of ${(x + y)^n}$ is $4096$. The greatest coefficient in the expansion is
Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3)\,\, ......\,\,$$(1 + x + x^2 + ..... + x^{30}) = $$a_0 + a_1x + a_2x^2$ .....$+$ $a_{465}x^{465}$, then sum of $a_0 + a_2 + a_4 + ......... +$ is
Value $\sum\limits_{r = 0}^{15} {\left( {{}^{15}{C_r}{}^{40}{C_{15}}{}^{20}{C_r} - {}^{35}{C_{15}}{}^{15}{C_r}{}^{25}{C_r}} \right)} $ is-
${C_1} + 2{C_2} + 3{C_3} + 4{C_4} + .... + n{C_n} = $
In the polynomial $(x - 1)(x - 2)(x - 3).............(x - 100),$ the coefficient of ${x^{99}}$ is