Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is.

Options

A. `(16)/(33)`
B. greater than `(1)/(3)`
C. less than `(12)/(33)`
D. `(10)/(33)`.

Answer

To solve the problem of finding the probability that exactly 2 pairs of gloves exist when 8 gloves are randomly selected from 6 pairs of distinct gloves, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 6 pairs of distinct gloves, which means there are a total of 12 gloves (6 left and 6 right). We need to select 8 gloves such that exactly 2 pairs are formed.

2. **Total Ways to Select 8 Gloves**:
The total number of ways to select 8 gloves from 12 is given by the combination formula:
\[ \text{Total ways} = \binom{12}{8} = \frac{12!}{8! \cdot 4!} \]
Calculating this:
\[ \binom{12}{8} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

3. **Favorable Cases for Exactly 2 Pairs**:
To have exactly 2 pairs, we first need to choose which 2 pairs will be included in our selection. The number of ways to choose 2 pairs from 6 is:
\[ \text{Ways to choose 2 pairs} = \binom{6}{2} = \frac{6!}{2! \cdot 4!} = 15 \]

4. **Selecting Remaining Gloves**:
After selecting 2 pairs (which gives us 4 gloves), we need to select 4 more gloves from the remaining 10 gloves (which consist of 4 left and 6 right gloves). However, we must ensure that these 4 gloves do not form any additional pairs.

- We can select 2 left gloves from the remaining 4 left gloves and 2 right gloves from the remaining 6 right gloves. The number of ways to do this is:
\[ \text{Ways to choose 2 left gloves} = \binom{4}{2} = 6 \]
\[ \text{Ways to choose 2 right gloves} = \binom{6}{2} = 15 \]
Therefore, the total ways to select these additional gloves is:
\[ 6 \times 15 = 90 \]

5. **Calculating Total Favorable Cases**:
The total number of favorable cases where exactly 2 pairs are formed is:
\[ \text{Total favorable cases} = \text{Ways to choose 2 pairs} \times \text{Ways to choose remaining gloves} = 15 \times 90 = 1350 \]

6. **Calculating the Probability**:
The probability of selecting 8 gloves such that exactly 2 pairs are formed is given by the ratio of favorable cases to total cases:
\[ P(\text{exactly 2 pairs}) = \frac{\text{Total favorable cases}}{\text{Total ways}} = \frac{1350}{495} \]
Simplifying this fraction:
\[ P(\text{exactly 2 pairs}) = \frac{1350 \div 45}{495 \div 45} = \frac{30}{11} \approx 2.727 \]

### Final Answer:
Thus, the probability that there exist exactly 2 pairs in the selection of 8 gloves is:
\[ P = \frac{30}{11} \approx 2.727 \]