What is the probability of picking up 2 red shirts from a pile of 15 shirts of which 7 are blue ? (only 2 colours shirts are there in it )

Options

A. `(1)/(4)`
B. `(4)/(15)`
C. `(2)/(15)`
D. `(3)/(8)`

Answer

To solve the problem of finding the probability of picking up 2 red shirts from a pile of 15 shirts (7 of which are blue), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Shirts**:
- We have a total of 15 shirts.

2. **Determine the Number of Blue Shirts**:
- The number of blue shirts is given as 7.

3. **Calculate the Number of Red Shirts**:
- Since there are only blue and red shirts, the number of red shirts can be calculated as:
\[ \text{Number of Red Shirts} = \text{Total Shirts} - \text{Blue Shirts} = 15 - 7 = 8 \]

4. **Define the Favorable Outcomes**:
- We need to find the number of ways to pick 2 red shirts from the 8 available red shirts. This can be calculated using the combination formula \( nCr \):
\[ \text{Favorable Outcomes} = \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \]

5. **Define the Total Outcomes**:
- The total number of ways to pick any 2 shirts from the 15 shirts is also calculated using the combination formula:
\[ \text{Total Outcomes} = \binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15 \times 14}{2 \times 1} = 105 \]

6. **Calculate the Probability**:
- The probability of picking 2 red shirts is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{2 Red Shirts}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{28}{105} \]

7. **Simplify the Probability**:
- We can simplify \( \frac{28}{105} \):
\[ \frac{28 \div 7}{105 \div 7} = \frac{4}{15} \]

### Final Answer:
The probability of picking up 2 red shirts from the pile is \( \frac{4}{15} \).

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