A survey of $450$ students about their subjects of interest resulted in the following outcome.

• $150$ students are interested in Mathematics.
• $200$ students are interested in Physics.
• $175$ students are interested in Chemistry.
• $50$ students are interested in Mathematics and Physics.
• $60$ students are interested in Physics and Chemistry.
• $40$ students are interested in Mathematics and Chemistry.
• $30$ students are interested in Mathematics, Physics, and Chemistry.
• Remaining students are interested in Humanities.

Based on the above information, the number of students interested in Humanities is

1. $10$
2. $30$
3. $40$
4. $45$

Given that,

• $n(\text{U}) = 450$
• $n(\text{M}) = 150$
• $n(\text{P}) = 200$
• $n(\text{C}) = 175$
• $n(\text{M} \cap \text{P}) = 50$
• $n(\text{P} \cap \text{C}) = 60$
• $n(\text{M} \cap \text{C}) = 40$
• $n(\text{M} \cap \text{P} \cap \text{C}) = 30$
• ${\color{Red}{n(\text{H}) = \;?}}$

Let’s draw the Venn diagram.

Now, $n(\text{M} \cup \text{P} \cup \text{C}) = n(\text{U}) + n(\text{M}) + n(\text{P}) – n(\text{C}) – n(\text{M} \cap \text{P}) – n(\text{P} \cap \text{C}) - n(\text{M} \cap \text{C}) + n(\text{M} \cap \text{P} \cap \text{C})$

$\Rightarrow n(\text{M} \cup \text{P} \cup \text{C}) = 150+200+175-50-60-40+30 = 555-150 = 405$

Now, $n(\text{U}) = n(\text{M} \cup \text{P} \cup \text{C}) + n(\text{H})$

$\Rightarrow n(\text{H}) = 450-405$

$\Rightarrow {\color{Blue}{\boxed{n(\text{H}) = 45}}}$

$\therefore$ The number of students interested in Humanities is $45.$

Correct Answer $:\text{D}$

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