here are 100 students in a class. In an examination, 50 of them
ailed in Mathematics, 45 failed in Physics, 40 failed in Biology
nd 32 failed in exactly two of three subjects. Only one studen
assed in all the subjects. Then the number of students failing
all the three subjects
a) Is 12
a) Is 4
t) Is 2
d) Cannot be determined from the given information

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UpStudy AI · Accepted Answer

Let's analyze the problem step by step.

**Given:**
- Total students = 100
- Failed in Mathematics (M) = 50
- Failed in Physics (P) = 45
- Failed in Biology (B) = 40
- Failed in exactly two subjects = 32
- Only one student passed all subjects (i.e., 99 failed at least one subject)

**Let:**
- $$ x $$ = Number of students who failed all three subjects.

**Using the principle of inclusion-exclusion:**

$$
\text{Total failing at least one} = F_M + F_P + F_B - (F_{MP} + F_{MB} + F_{PB}) + F_{MPB}
$$

Where:
- $$ F_{MP} $$, $$ F_{MB} $$, $$ F_{PB} $$ = Students failing exactly two subjects.
- $$ F_{MPB} = x $$ (students failing all three).

Given that:
$$
F_{MP} + F_{MB} + F_{PB} = 32 + 3x
$$

Plugging in the values:
$$
99 = 50 + 45 + 40 - (32 + 3x) + x
$$
$$
99 = 135 - 32 - 2x
$$
$$
99 = 103 - 2x
$$
$$
2x = 4 \implies x = 2
$$

**Answer:** 
**2**
2 students failed all three subjects.

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