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In the examination of the class of 100 students,
50 students failed in Mathematics, 45 failed in Physics, 40 failed in Biology
and 32 failed in exactly two of the three subjects mentioned.
If only one student passed in all subjects then find the number of the students that failed in three subjects?
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This problem is on Inclusion-Exclusion principle.
We have 3 subsets M, P and B with the given cardinality of each.
Let MP, MB and PB be their in-pairs intersection, and MPC be their triple intersection.
The number of elements in MPC is unknown; let denote this unknown as x.
From the problem, we have this condition (this equation)
| MP \ MPB | + | MB \ MPB | + | PB \ MPB | = 32,
or, which is the same
(MP -x) + (MB - x) + (PB - x) = 32,
or
MP + MB + PB = 32 + 3x.
Now I write the inclusion-exclusion principle equation for the union M U P U B
M U P U B = M + P + B - MP - MB - PB + MPB.
Next, I replace M U P U B by 100-1 = 99; replace M, P and B by the given numbers; and replace MP + MB + PB by 32 + 3x. I get then
99 = 50 + 45 + 40 - (32+3x) + x.
It gives
99 = 50 + 45 + 40 - 32 - 2x
2x = 50 + 45 + 40 - 32 - 99
2x = 4
x = 4/2 = 2.
ANSWER. The number of the students that failed in three subjects is 2.
Solved.