Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be two events such that $$P(B \mid A)=\frac{2}{5}, P(A \mid B)=\frac{1}{7}$$ and $$P(A \cap B)=\frac{1}{9} \cdot$$ Consider

(S1) $$P\left(A^{\prime} \cup B\right)=\frac{5}{6}$$,

(S2) $$P\left(A^{\prime} \cap B^{\prime}\right)=\frac{1}{18}$$

Then :

Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}$$ and $$\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0$$, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then $$50-\frac{2 d}{\beta^{2}}$$ is equal to __________.

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then $$\mathrm{b}+3 \mathrm{~g}$$ is equal to ____________.

Let $$\mathrm{z}=a+i b, b \neq 0$$ be complex numbers satisfying $$z^{2}=\bar{z} \cdot 2^{1-z}$$. Then the least value of $$n \in N$$, such that $$z^{n}=(z+1)^{n}$$, is equal to __________.