The probability of Jide, Atu and Obu solving a given problem are $\frac{1}{12}$ , $\frac{1}{6}$ and $\frac{1}{8}$ respectively. Calculate the probability that only one solves the problem.

$\frac{1}{576}$

$\frac{55}{576}$

$\frac{77}{576}$

$\frac{167}{576}$

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Correct answer is D

$P(Jide) = \frac{1}{12}; P(\text{not Jide}) = \frac{11}{12}$

$P(Atu) = \frac{1}{6}; P(\text{not Atu}) = \frac{5}{6}$

$P(Obu) = \frac{1}{8}; P(\text{not Obu}) = \frac{7}{8}$

$P(\text{only one of them}) = P(\text{Jide not Atu not Obu}) + P(\text{Atu not Jide not Obu}) + P(\text{Obu not Jide not Atu})$

= $(\frac{1}{12} \times \frac{5}{6} \times \frac{7}{8}) + (\frac{1}{6} \times \frac{11}{12} \times \frac{7}{8}) + (\frac{1}{8} \times \frac{11}{12} \times \frac{5}{6})$

= $\frac{35}{576} + \frac{77}{576} + \frac{55}{576}$

= $\frac{167}{576}$

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The probability of Jide, Atu and Obu solving a given problem are $\frac{1}{12}$ , $\frac{1}{6}$ and $\frac{1}{8}$ respectively. Calculate the probability that only one solves the problem.