Answer

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Hint: -You have probability of solving problems by students A, B and C is given you have find probability that problem will be solved will be equal to 1- probability of problem can’t be solved.

Complete step-by-step answer:

Given

$P\left( A \right) = \dfrac{1}{2},P\left( B \right) = \dfrac{3}{4},P\left( C \right) = \dfrac{1}{4}$

$

\Rightarrow P\left( {\vec A} \right) = 1 - P\left( A \right) \\

\therefore P\left( {\vec A} \right) = 1 - \dfrac{1}{2} = \dfrac{1}{2} \\

$

$

\Rightarrow P\left( {\vec B} \right) = 1 - P\left( B \right) \\

\Rightarrow P\left( {\vec B} \right) = 1 - \dfrac{3}{4} = \dfrac{1}{4} \\

$

$ \Rightarrow P\left( {\vec C} \right) = 1 - P\left( C \right)$

$\therefore P\left( {\vec C} \right) = 1 - \dfrac{1}{4} = \dfrac{1}{4}$

The probability that problem will be solved = 1- (probability that problem can’t be solved).

$\therefore $Required probability = 1- $P\left( {\vec A \cap \vec B \cap \vec C} \right)$

$\therefore 1 - \dfrac{1}{2} \times \dfrac{1}{4} \times \dfrac{3}{4} = \dfrac{{29}}{{32}}$

Hence option C is the correct option.

Note: -Whenever you get this type of question the key concept of solving is if you have to find probability of success you can write it 1- probability of failure. And you have knowledge of $P\left( {\vec A \cap \vec B \cap \vec C} \right) = P\left( {\vec A} \right) \times P\left( {\vec B} \right) \times P\left( {\vec C} \right)$.

Complete step-by-step answer:

Given

$P\left( A \right) = \dfrac{1}{2},P\left( B \right) = \dfrac{3}{4},P\left( C \right) = \dfrac{1}{4}$

$

\Rightarrow P\left( {\vec A} \right) = 1 - P\left( A \right) \\

\therefore P\left( {\vec A} \right) = 1 - \dfrac{1}{2} = \dfrac{1}{2} \\

$

$

\Rightarrow P\left( {\vec B} \right) = 1 - P\left( B \right) \\

\Rightarrow P\left( {\vec B} \right) = 1 - \dfrac{3}{4} = \dfrac{1}{4} \\

$

$ \Rightarrow P\left( {\vec C} \right) = 1 - P\left( C \right)$

$\therefore P\left( {\vec C} \right) = 1 - \dfrac{1}{4} = \dfrac{1}{4}$

The probability that problem will be solved = 1- (probability that problem can’t be solved).

$\therefore $Required probability = 1- $P\left( {\vec A \cap \vec B \cap \vec C} \right)$

$\therefore 1 - \dfrac{1}{2} \times \dfrac{1}{4} \times \dfrac{3}{4} = \dfrac{{29}}{{32}}$

Hence option C is the correct option.

Note: -Whenever you get this type of question the key concept of solving is if you have to find probability of success you can write it 1- probability of failure. And you have knowledge of $P\left( {\vec A \cap \vec B \cap \vec C} \right) = P\left( {\vec A} \right) \times P\left( {\vec B} \right) \times P\left( {\vec C} \right)$.

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