For an examination a candidate has to select 7 questions from three different groups A , B and C. The three groups contain 4, 5 and 6 questions respectively. In how many different ways can a candidate make his selection if he has to select atleast 2 questions from each group?

Step-by-step solution

The candidate must select 7 questions from groups A, B, and C containing 4, 5, and 6 questions respectively, with at least 2 questions from each group. Let n_A, n_B, n_C be the number of questions selected from groups A, B, and C respectively. The conditions are n_A + n_B + n_C = 7 and n_A 2, n_B 2, n_C 2 . The possible integer solutions (n_A, n_B, n_C) are: 1. (3, 2, 2) : ⁴C₃ ⁵C₂ ⁶C₂ = 4 10 15 = 600 2. (2, 3, 2) : ⁴C₂ ⁵C₃ ⁶C₂ = 6 10 15 = 900 3. (2, 2, 3) : ⁴C₂ ⁵C₂ ⁶C₃ = 6 10 20 = 1200 Summing these cases: 600 + 90