If $(1 -x + x^2)^n = a_0 + a_1x + a_2x^2 + ……. + a_{2n}x^{2n}$, then $a_0 + a_2 + a_4 +……..+ a_{2n}$ is equal to

If ${C_r}$ stands for $^n{C_r}$, the sum of the given series $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 – 2C_1^2 + 3C_2^2 – ….. + {( – 1)^n}(n + 1)C_n^2]$, Where $n$ is an even positive integer, is

The sum of the co-efficients of all odd degree terms in the expansion of  ${\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},\left( {x > 1} \right)$ 

$(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) …… (1 + x + x^2 + …… + x^{100})$ when written in the ascending power of $x$ then the highest exponent of $x$ is ______ .

For integers $n$ and $r$, let $\left(\begin{array}{l} n \\ r \end{array}\right)=\left\{\begin{array}{ll}{ }^{n} C _{ r }, & \text { if } n \geq r \geq 0 \\ 0, & \text { otherwise }\end{array}\right.$

The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}15 \\ k-i\end{array}\right)+\sum_{i=0}^{k+1}\left(\begin{array}{c}12 \\ i\end{array}\right)\left(\begin{array}{c}13 \\ k+1-i\end{array}\right)$ exists, is equal to …… .