Question

In a class of $55$ students, the number of students studying different subjects is $23$ in Mathematics, $24$ in physics, $19$ in chemistry, $12$ in Mathematics and Physics, $9$ in Mathematics and chemistry, $7$ in physics and chemistry and $4$ in all the three subjects.

The number of students who have taken exactly one the subjects is

• A. $6$
• B. $9$
• C. $7$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Step 1: Find the total number of students:

Let the total number of students be $\mathrm{N}$. Then $\mathrm{N}$$=55$

Let, $\mathrm{n}\left(\mathrm{M}\right)=$ students who are studying mathematics, $\mathrm{n}\left(\mathrm{C}\right)=$students who are studying chemistry, $\mathrm{n}\left(\mathrm{P}\right)=$students who are studying Physics.

$\begin{array}{rcl}\mathrm{n}\left(\mathrm{M}\right)& =& 23,\\ \mathrm{n}\left(\mathrm{P}\right)& =& 24,\\ \mathrm{n}\left(\mathrm{C}\right)& =& 19,\\ \mathrm{n}(\mathrm{M}\cap \mathrm{P})& =& 12,\\ \mathrm{n}(\mathrm{P}\cap \mathrm{C})& =& 7,\\ \mathrm{n}(\mathrm{M}\cap \mathrm{C})& =& 9,\\ \mathrm{n}(\mathrm{M}\cap \mathrm{P}\cap \mathrm{C})& =& 4\end{array}$

Step 2: Find the number of students studying only mathematics:

Now, the number of students studying only Mathematics is,

$\mathrm{n}(\mathrm{M}\cap \mathrm{P}\text{'}\cap \mathrm{C}\text{'})=\mathrm{n}\left(\mathrm{M}\right)-\mathrm{n}(\mathrm{M}\cap \mathrm{P})-\mathrm{n}(\mathrm{M}\cap \mathrm{C})+\mathrm{n}(\mathrm{M}\cap \mathrm{P}\cap \mathrm{C})$

Substituting the values, we get

$\mathrm{n}\left(\mathrm{M}\cap \mathrm{P}\text{'}\cap \mathrm{C}\text{'}\right)=23-9-12+4$

$=6$

Step 3: Find the number of students studying only Physics:

Now, the number of students studying only Physics is,

$\mathrm{n}(\mathrm{P}\cap \mathrm{M}\text{'}\cap \mathrm{C}\text{'})=\mathrm{n}\left(\mathrm{P}\right)-(\mathrm{P}\cap \mathrm{M})-(\mathrm{P}\cap \mathrm{C})+\mathrm{n}(\mathrm{P}\cap \mathrm{M}\cap \mathrm{C})$

$=24-12-7+4$

$=9$

Step 4: Find the number of students studying only Chemistry:

Now, the number of students studying only Chemistry is,

$\mathrm{n}(\mathrm{C}\cap \mathrm{M}\text{'}\cap \mathrm{P}\text{'})=\mathrm{n}\left(\mathrm{C}\right)-\mathrm{n}(\mathrm{C}\cap \mathrm{M})-\mathrm{n}(\mathrm{C}\cap \mathrm{P})+\mathrm{n}(\mathrm{M}\cap \mathrm{P}\cap \mathrm{C})$

$$\begin{gathered} =19-9-7+4 \\ =7 \end{gathered}$$

Step 5: Finding the number of students taking only one subject:

Therefore the number of people who have taken exactly one of the three subjects

$$\begin{gathered} =6+9+7 \\ =22 \end{gathered}$$

Hence, the correct option is D.