$^5C_0+2.^5C_1+2^2.^5C_2+2^3.^5C_3 +2^4.^5C_4+2^5.^5C_5 =$

Number of different terms in the sum $( 1 + x ) ^ { 2009 } \cdot \left( 1 + x ^ { 2 } \right) ^ { 2008 } + \left( 1 + x ^ { 3 } \right) ^ { 2007 } ,$ is

Find the value of $\dfrac{1}{\left(n-1\right)!}+\dfrac{1}{\left(n-3\right)!3!}+\dfrac{1}{\left(n-5\right)!5!}+...$

If $T_r=^{2016}C_rx^{2016-r}$, for $r=0, 1, ,....2016$, then $(T_0 - T_2+T_4....+T_{2016})^2+(T_1-T_3+T_5....T_{2015})^2$ is equal to - ;

$(1+x)^{15}=a_0+a_1x+\ldots\ldots+a_{15}x^{15} \Rightarrow \sum_{r=1}^{15}r\frac{a_r}{a_{r-1}}=$