A company has 6 software engineers, 5 data scientists, and 4 product managers. A committee of 10 members is to be formed such that it includes at least 2 members from each of the three professions. The total number of ways to form the committee is _________.

Step-by-step solution

Total number of people = 6 + 5 + 4 = 15 . Total unrestricted ways to select 10 members from 15 people is: ¹⁵C₁₀ = ¹⁵C₅ = 15 14 13 12 11 5 4 3 2 1 = 3003 We must subtract the cases where the minimum constraint (at least 2 from each profession) is violated. Case 1: Less than 2 software engineers are selected. Number of ways to select 1 software engineer = ⁶C₁ ⁹C₉ = 6 Number of ways to select 0 software engineers = 0 (since we need 10 members but only 9 others are available) Total violations for software engineers = 6