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 probability as ratio of favorable outcomes (exactly 2 complete pairs from 6 pairs with 4 remaining gloves unpaired) to total outcomes (8 gloves selected from 12). Use combinations: favorable = C(6,2) × 2^4 (choosing 2 pairs and distributing 4 single gloves from remaining 4 pairs) divided by C(12,8).
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