The range of a random variable X= {1,2,3....} and the probabilities are given by $\mathrm{P}(X=\mathrm{k})=\frac{3^{CK}}{\mathrm{K}!}$ (K=1,2,3.....) and C is a constant. Then C = 

 

Given  $\mathrm{P}(X=\mathrm{k})=\frac{3^{\mathrm{CK}}}{\mathrm{K}!}, \mathrm{k}=1, 2, 3, 4, .....$

Since  $\sum _{\mathrm{k}=1}^{\infty } \mathrm{P}(X=\mathrm{K})=1$

      $$\begin{gathered} \Rightarrow \sum _{\mathrm{k}=1}^{\infty } \frac{3^{\mathrm{CK}}}{\mathrm{K}!}=1 \\ \Rightarrow \frac{3^{\mathrm{C}}}{1!}+\frac{3^{2\mathrm{C}}}{2!}+\frac{3^{3\mathrm{C}}}{3!}+......=1 \\ \Rightarrow 1+\frac{{\left(3^{\mathrm{C}}\right)}^1}{1!}+\frac{{\left(3^{\mathrm{C}}\right)}^2}{2!}+\frac{{\left(3^{\mathrm{C}}\right)}^3}{3!}+.......=1+1 \\ \Rightarrow {\mathrm{e}}^{3^C}=2 \\ \Rightarrow 3^{\mathrm{C}}={\log }_{e}2 \\ \Rightarrow \mathrm{C}={\log }_3 \left({\log }_{e}2\right) \end{gathered}$$