There are 10 different pairs of shoes in a wardrobe. Find the probability of the event that there is at least one pair among them in choosing 4 shoes by chance. Express your answer in rational form.

3/5

0.4

To find the probability of there being at least one pair of shoes among the 4 shoes chosen, we need to calculate the complement of the event that no pairs are chosen.

First, let's determine the total number of possible combinations when selecting 4 shoes out of the 10 pairs. This can be calculated using the formula for combinations, which is nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items chosen. In this case, n = 10 pairs * 2 shoes per pair = 20 shoes and r = 4 shoes.

So, the total number of combinations is 20C4 = 20! / (4!(20-4)!) = 4845.

Now, let's calculate the number of combinations where no pairs are chosen. Since there are 10 pairs, we need to choose 4 individual shoes from the remaining 20 - 10 = 10 individual shoes.

So, the number of combinations with no pairs chosen is 10C4 = 10! / (4!(10-4)!) = 210.

Since we're interested in the probability of the complement of this event, the probability of no pairs chosen is 210 / 4845.

Now, to find the probability of at least one pair being chosen, we subtract the probability of no pairs chosen from 1 (since the two events are complements).

The probability of at least one pair among the 4 shoes is 1 - 210 / 4845 = 4635 / 4845 = 463 / 484.

Therefore, the probability is 463/484 in rational form.