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

$P(x)=\left(2+\frac{x}{4}\right)^{11}, q(x)=(1+c x)^{n}, n \in N, c$ ধ্রবক।

Let $s_n = 1 + q + q^2 + ................. + q^n$ and

$T_n = 1 + \left(\dfrac{q + 1}{2}\right) + \left(\dfrac{q + 1}{2}\right)^2 +......... \left(\dfrac{q + 1}{2}\right)^n$

where q is a real number and q $\neq$ 1.

If $^{101}C_1 + ^{101}{C_2}.S_{1} +...... + ^{101}C_{101}.S_{100} = \alpha T_{100}, then\,\,\, \alpha$ is equal to :-