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, we need to determine how many students opted for neither NCC nor NSS and then calculate the probability of selecting one of those students at random.

### Step-by-Step Solution:

1. **Identify the given data:**
- Total number of students (T) = 60
- Students who opted for NCC (NCC) = 40
- Students who opted for NSS (NSS) = 30
- Students who opted for both NCC and NSS (Both) = 20

2. **Use the principle of inclusion-exclusion to find the number of students who opted for either NCC or NSS or both:**
\[ \text{Number of students opting for either NCC or NSS} = \text{NCC} + \text{NSS} - \text{Both} \]
\[ = 40 + 30 - 20 = 50 \]

3. **Calculate the number of students who opted for neither NCC nor NSS:**
\[ \text{Students opting for neither} = \text{Total students} - \text{Students opting for either NCC or NSS} \]
\[ = 60 - 50 = 10 \]

4. **Calculate the probability that a randomly selected student opted for neither NCC nor NSS:**
\[ \text{Probability} = \frac{\text{Number of students opting for neither}}{\text{Total number of students}} \]
\[ = \frac{10}{60} = \frac{1}{6} \]

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

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