Murari has 9 pairs of dark blue socks and 9 pairs of black socks. He keeps them all in the same bag. If he picks out three socks at random, what is the probability that he will get a matching pair?

Options

A. `(""^(9)C_(3) xx ""^(9)C_1)/(""^(18)C_3)`
B. `(2 xx ""^(9)C_(3) xx ""^(9)C_1)/(""^(18)C_3)`
C. 1
D. `4/7`

Answer

To solve the problem, we need to determine the probability that Murari will get at least one matching pair of socks when he randomly picks three socks from a bag containing 9 pairs of dark blue socks and 9 pairs of black socks.

### Step-by-Step Solution:

1. **Identify the Total Number of Socks**:
- Murari has 9 pairs of dark blue socks, which means he has \(9 \times 2 = 18\) dark blue socks.
- He also has 9 pairs of black socks, which means he has \(9 \times 2 = 18\) black socks.
- Therefore, the total number of socks is \(18 + 18 = 36\) socks.

2. **Understanding the Selection**:
- When Murari picks 3 socks at random from these 36 socks, we need to analyze the possible outcomes.
- There are only two types of socks: dark blue and black.

3. **Analyzing the Possible Outcomes**:
- When picking 3 socks, there are only two types of socks available. This means that at least two of the socks must be of the same type.
- The possible combinations of socks when picking 3 are:
- 3 dark blue socks
- 3 black socks
- 2 dark blue socks and 1 black sock
- 1 dark blue sock and 2 black socks

4. **Conclusion**:
- In all possible combinations of picking 3 socks, at least 2 socks will always be of the same type (either dark blue or black).
- Therefore, it is certain that Murari will get at least one matching pair.

5. **Calculating the Probability**:
- Since it is certain that he will get a matching pair, the probability of getting at least one matching pair is \(1\).

### Final Answer:
The probability that Murari will get a matching pair when he picks 3 socks at random is \(1\).

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