A task force of 10 people is to be selected from 3 departments of a company: HR ( 6 people), IT ( 5 people), and Sales ( 4 people). The task force must have at least 2 people from each department. However, Mr. A (from HR) and Mr. B (from IT) refuse to work together on the task force. The total number of valid ways to form the task force is _________.

Step-by-step solution

Total valid ways to form the task force without the refusal constraint can be found using complementary counting. Total unrestricted ways to select 10 from 15 people = ¹⁵C₁₀ = 3003 . Violations for HR ( Violations for IT ( Violations for Sales ( Since there are no overlaps between these violations, total valid ways without refusal constraint = 3003 - (6 + 51 + 231) = 2715 . Now, we find the number of valid ways where both Mr. A and Mr. B are selected. If both are selected, we need 8 more people from the remaining 1