In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is

Options

A. `(2)/(3)`
B. `(1)/(6)`
C. `(1)/(3)`
D. `(5)/(6)`

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the number of students who opted for neither NCC nor NSS.

### Step 1: Identify the total number of students
We know that the total number of students in the class is:
\[ \text{Total students} = 60 \]

### Step 2: Identify the number of students who opted for NCC, NSS, and both
- Number of students who opted for NCC (NCC) = 40
- Number of students who opted for NSS (NSS) = 30
- Number of students who opted for both NCC and NSS = 20

### Step 3: Apply the principle of inclusion-exclusion
To find the number of students who opted for at least one of the activities (NCC or NSS), we use the formula:
\[ \text{NCC} \cup \text{NSS} = \text{NCC} + \text{NSS} - \text{NCC} \cap \text{NSS} \]
Substituting the values:
\[ \text{NCC} \cup \text{NSS} = 40 + 30 - 20 = 50 \]

### Step 4: Calculate the number of students who opted for neither NCC nor NSS
To find the number of students who opted for neither activity, we subtract the number of students who opted for at least one activity from the total number of students:
\[ \text{Neither NCC nor NSS} = \text{Total students} - (\text{NCC} \cup \text{NSS}) \]
Substituting the values:
\[ \text{Neither NCC nor NSS} = 60 - 50 = 10 \]

### Step 5: Calculate the probability that a randomly selected student opted for neither NCC nor NSS
The probability \( P \) that a randomly selected student opted for neither NCC nor NSS is given by:
\[ P(\text{Neither NCC nor NSS}) = \frac{\text{Number of students who opted for neither}}{\text{Total number of students}} = \frac{10}{60} = \frac{1}{6} \]

### Final Answer
The probability that the student selected has opted neither for NCC nor for NSS is:
\[ \frac{1}{6} \]