Answer:
185/198
Explanation:
Since the are 12 Cd and we require them to be selected in 5 ways, the number of selections is ¹²C₅ = 12 × 11 × 10 × 9 × 8/5! = 95040/120 = 792 ways
Since we have Four are rap music, five are country music, and three are heavy metal music, the number of ways to select them in groups of five so as to have at least one of each CD from each category, we have
For the first selection 3 of one category, and 1 each of the other categories.
The second selection is 2 each of two categories and 1 of the last category.
So, doing this for each category, we have
⁴C₃ × ⁵C₁ × ³C₁ + ⁴C₂ × ⁵C₂ × ³C₁ + ³C₃ × ⁴C₁ × ⁵C₁ + ³C₂ × ⁴C₂ × ⁵C₁ + ⁵C₃ × ⁴C₁ × ³C₁ + ⁵C₂ × ³C₂ × ⁴C₁
= (4 × 3 × 2/3! × 5 × 3) + (4 × 3/2! × 5 × 4/2! × 3) + (1 × 4 × 5) + (3 × 2/2! × 4 × 3/2! × 5) + (5 × 4 × 3/2! × 4 × 3) + (5 × 4/2! × 3 × 2/2! × 4)
= 360/6 + 720/4 + 20 + 360/4 + 720/2 + 120/4
= 60 + 180 + 20 + 90 + 360 + 30
= 740 ways
So, the required probability is P = number of ways of selecting at least one CD/ total number of ways = 740/792 = 185/198