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 of finding the probability that there exist exactly 2 pairs of gloves when selecting 8 gloves from 6 pairs of distinct gloves, we can follow these steps:

### Step 1: Calculate the Total Number of Ways to Select 8 Gloves from 12
We start by calculating the total number of ways to choose 8 gloves from the 12 available gloves (6 pairs). This can be represented mathematically as:

\[ \text{Total Ways} = \binom{12}{8} = \binom{12}{4} \]

Using the formula for combinations:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

We can calculate:

\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495 \]