In a group of students , there are 3 boys and 3 girls . Four students are to be selected at random from the group . Find the probability that either 3 boys and 1 girl or 3 girls and 1 boy are selected .

Answer

To solve the problem, we need to find the probability of selecting either 3 boys and 1 girl or 3 girls and 1 boy from a group of 3 boys and 3 girls.

### Step-by-step Solution:

1. **Identify the total number of students:**
- There are 3 boys and 3 girls, making a total of \(3 + 3 = 6\) students.

2. **Calculate the total number of ways to select 4 students from 6:**
- The total number of ways to choose 4 students from 6 is given by the combination formula \(C(n, r) = \frac{n!}{r!(n-r)!}\).
- Here, \(n = 6\) and \(r = 4\):
\[ C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} = \frac{6 \times 5}{2 \times 1} = 15 \]

3. **Calculate the number of ways to select 3 boys and 1 girl:**
- The number of ways to choose 3 boys from 3 is \(C(3, 3)\) and to choose 1 girl from 3 is \(C(3, 1)\):
\[ C(3, 3) = 1 \quad \text{and} \quad C(3, 1) = 3 \]
- Therefore, the total ways to select 3 boys and 1 girl:
\[ C(3, 3) \times C(3, 1) = 1 \times 3 = 3 \]

4. **Calculate the number of ways to select 3 girls and 1 boy:**
- The number of ways to choose 3 girls from 3 is \(C(3, 3)\) and to choose 1 boy from 3 is \(C(3, 1)\):
\[ C(3, 3) = 1 \quad \text{and} \quad C(3, 1) = 3 \]
- Therefore, the total ways to select 3 girls and 1 boy:
\[ C(3, 3) \times C(3, 1) = 1 \times 3 = 3 \]

5. **Calculate the total favorable outcomes:**
- The total number of ways to select either 3 boys and 1 girl or 3 girls and 1 boy:
\[ \text{Total favorable outcomes} = 3 + 3 = 6 \]

6. **Calculate the probability:**
- The probability of selecting either 3 boys and 1 girl or 3 girls and 1 boy is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{3 boys and 1 girl or 3 girls and 1 boy}) = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{6}{15} = \frac{2}{5} \]

### Final Answer:
The probability that either 3 boys and 1 girl or 3 girls and 1 boy are selected is \(\frac{2}{5}\).