Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is :
Step-by-Step Solution
Key Concept: Calculate favorable outcomes by selecting exactly 2 complete pairs from 6 pairs and 4 individual gloves from remaining 4 pairs such that no additional complete pair is formed. Use combinatorics: C(6,2) ways to choose 2 pairs, then C(4,4)×2^4 ways to select one glove from each of 4 remaining pairs, divided by total C(12,8) ways to select 8 gloves from 12.
Favorable cases where all 6 balls are distributed among at least 3 boxes with at least one ball in each: $m = ^6C_4(2^4) + 2^6C_1(^6C_1) + ^6C_2(^6C_2) = 16$. Total cases: $n = ^{12}C_6$. The probability is $ rac{m}{n} = rac{16}{33}$.
Correct Answer: 1