Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is a/b where a and b are co-prime then a is______.

Answer

To solve the problem, we need to find the probability that out of 6 pairs of distinct gloves, when 8 gloves are randomly selected, there exist exactly 2 pairs. Let's break this down step by step.

### Step 1: Understanding the Total Number of Gloves
We have 6 pairs of distinct gloves, which means there are a total of:
\[ 12 \text{ gloves} \quad (6 \text{ pairs} \times 2 \text{ gloves per pair}) \]

### Step 2: Total Ways to Choose 8 Gloves
We need to calculate the total number of ways to choose 8 gloves from the 12 available gloves. This can be calculated using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):
\[ \text{Total ways} = \binom{12}{8} = \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]

### Step 3: Selecting Exactly 2 Pairs
To have exactly 2 pairs in the selection of 8 gloves, we need to:
1. Choose 2 pairs from the 6 available pairs.
2. Choose 4 additional gloves from the remaining 4 pairs (ensuring that we do not complete any additional pairs).

#### Step 3.1: Choosing 2 Pairs
The number of ways to choose 2 pairs from 6 pairs is:
\[ \text{Ways to choose 2 pairs} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]

#### Step 3.2: Choosing 4 Gloves from Remaining 4 Pairs
After selecting 2 pairs, we have 4 pairs left. We need to choose 4 gloves from these 4 pairs, ensuring that we only select one glove from each pair. The number of ways to choose 1 glove from each of the 4 remaining pairs is:
\[ \text{Ways to choose 4 gloves} = 2^4 = 16 \]
(Each pair contributes 2 choices: either the left glove or the right glove.)

### Step 4: Total Favorable Outcomes
Now, we can calculate the total number of favorable outcomes for having exactly 2 pairs:
\[ \text{Total favorable outcomes} = \binom{6}{2} \times 2^4 = 15 \times 16 = 240 \]

### Step 5: Calculating the Probability
Now, we can find the probability of selecting exactly 2 pairs:
\[ \text{Probability} = \frac{\text{Total favorable outcomes}}{\text{Total ways to choose 8 gloves}} = \frac{240}{495} \]

### Step 6: Simplifying the Probability
To simplify \( \frac{240}{495} \), we can find the greatest common divisor (GCD) of 240 and 495. The GCD is 15:
\[ \frac{240 \div 15}{495 \div 15} = \frac{16}{33} \]

### Final Step: Identifying \( a \) and \( b \)
Here, we have the probability in the form \( \frac{a}{b} \) where \( a = 16 \) and \( b = 33 \). Since 16 and 33 are co-prime, we conclude that:
\[ \text{The value of } a = 16 \]