Question

Out of 180 students in a class, 7 students failed in all the three subjects viz, Physics, Chemistry, Maths. Only 144 students pass in only one subject and 21 students pass in only two subjects.
(a) How many students passed in all the three subjects?
(b) How many students passed in at least two subjects?
(c) How many students passed in at most two subjects but at least in one subject?

• A. 8, 25, 150
• B. 8, 29, 145
• C. 8, 29, 165
• D. 8, 25, 130

Answer 1

The correct option is C 8, 29, 165

Given, Number of students passing in at least one subject = X = ( I + II + III) - Number of students failing in all the subjects = 180 - 7 = 173 - - - (i)
I = 144 - - - (ii)
II = 21 - - - - (iii)
144 + 21 + III = 173 $\Rightarrow$ III = 8
Number of students passed in all three subjects = III = 8
Number of students passed in at least two subjects = II + III = 21 + 8 = 29
Number of students passed in at most two subjects but at least in one subject = I + II = 144 + 21 = 165

Alternatively,

$\alpha =a+b+c=144\beta =x+y+z=21\alpha +\beta +\gamma =180-7=173\gamma =(\alpha +\beta +\gamma )-(\alpha +\beta )\gamma =\left(173\right)-(144+21)\Rightarrow \gamma =8(\gamma =k)\left(a\right)8$
(b) Atleast two subjects
$=\beta +\gamma =21+8=29$
(c) Atleast one subject but
atmost two subjects
$=\alpha +\beta =144+21=165.$