One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

$5 \text{ couples (1 Indian} + 4 \text{ American)} = 10 \text{ people}$

$\text{seated randomly around a circular table}$

$\text{To find: } P(\text{Indian man beside wife} \mid \text{each American man beside wife})$

$\text{Condition: each American couple is adjacent}$

$\Rightarrow \text{ treat each American couple as ONE block}$

$4 \text{ American blocks} + \text{Indian man} + \text{Indian woman} = 6 \text{ objects}$

$\text{want: the two Indian singles adjacent}$

$n \text{ distinct objects on a circle} \Rightarrow (n-1)! \text{ arrangements}$

$N = (6-1)! \cdot 2^4 = 5! \cdot 2^4$

$N = 120 \cdot 16 = 1920$

$\text{Indian couple together} \Rightarrow 5 \text{ blocks in a circle} = 4! \text{ ways}$

$M = 4! \cdot 2^4 \cdot 2$

$M = 24 \cdot 16 \cdot 2 = 768$

$P = \dfrac{M}{N} = \dfrac{768}{1920}$

$P = \dfrac{4! \cdot 2^4 \cdot 2}{5! \cdot 2^4} = \dfrac{4! \cdot 2}{5!} = \dfrac{24 \cdot 2}{120} = \dfrac{2}{5}$

$P = \dfrac{2}{5}$

$\boxed{\text{Answer: (3)}\ \ \dfrac{2}{5}}$