The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. Of these subjects, the student has a 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40% chance of passing in exactly two. Which of the following relations are true ?

A
$p + m + c = \frac{19}{20}$
B
$p + m + c = \frac{27}{20}$
C
$pmc = \frac{1}{10}$
D
$pmc = \frac{1}{4}$

Solution:

$\begin{array}{l} P (m \cup p \cup c) = \frac{75}{100} \\ \Rightarrow m + p + c - \sum mp + mpc = \frac{3}{4} \to (1) \end{array}$
p(passing at least 2 subjects)
$= \sum mp - 2 mpc = \frac{50}{100} = \frac{1}{2} \to (2)$
p(passing exactly 2 subjects)
$= \sum mp - 3 mpc = \frac{40}{100} = \frac{2}{5} \to (3)$
(2) and (3) gives $mpc = \frac{1}{2} - \frac{2}{5} = \frac{1}{10}$
From (2) $\sum mp = \frac{1}{2} + 2 mpc = \frac{1}{2} + \frac{1}{5} = \frac{7}{10}$
From $\sum m = \frac{3}{4} + \sum mp - mpc$
$= \frac{3}{4} + \frac{7}{10} - \frac{1}{10} = \frac{15 + 14 - 2}{20} = \frac{27}{20}$

Solution:

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