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
$\frac{{(2n)!}}{{(n + 1)!(n + 2)!}}$
$\frac{{(2n)!}}{{(n - 2)!(n + 2)!}}$
$\frac{{(2n)!}}{{(n)!(n + 2)!}}$
$\frac{{(2n)!}}{{(n - 1)!(n + 2)!}}$
Let $\left( a + bx + cx ^2\right)^{10}=\sum \limits_{ i =0}^{20} p _{ i } x ^{ i }, a , b , c \in N$. If $p _1=20$ and $p _2=210$, then $2( a + b + c )$ is equal to
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
The sum of last eight consecutive coefficients in the expansion of $(1+x)^{15}$ 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 ...... .
Coefficient of $x^{n-6}$ in the expansion $n\left[ {x - \left( {\frac{{^n{C_0}{ + ^n}{C_1}}}{{^n{C_0}}}} \right)} \right]\left[ {\frac{x}{2} - \left( {\frac{{^n{C_1}{ + ^n}{C_2}}}{{^n{C_1}}}} \right)} \right]\left[ {\frac{x}{3} - \left( {\frac{{^n{C_2}{ + ^n}{C_3}}}{{^n{C_2}}}} \right)} \right].....$ $ \left[ {\frac{x}{n} - \left( {\frac{{^n{C_{n - 1}}{ + ^n}{C_n}}}{{^n{C_{n - 1}}}}} \right)} \right]$ is equal to (where $n = n . (n -1) . (n -2).... 3.2.1$ )