Group P consists of 5 Doctors and 4 Engineers, while Group Q consists of 4 Doctors and 5 Engineers. A committee of 4 Doctors and 4 Engineers is to be formed such that exactly 4 members are from Group P and 4 members are from Group Q. If a specific Doctor from Group P and a specific Engineer from Group Q refuse to serve on the committee together, then the number of valid ways to form the committee is :

Step-by-step solution

First, we calculate the total number of unrestricted ways to form the committee. Let the number of Doctors and Engineers selected from Group P be d and e . Then the number of Doctors and Engineers selected from Group Q must be 4-d and 4-e . Since 4 members are from Group P, d + e = 4 . The possible cases for (d, e) from Group P and the corresponding selections from Group Q are: Case 1: P(4D, 0E) and Q(0D, 4E) (⁵C₄ ⁴C₀) (⁴C₀ ⁵C₄) = 5 1 1 5 = 25 Case 2: P(3D, 1E) and Q(1D, 3E) (⁵C₃ ⁴C₁) (⁴C₁ ⁵C₃) = 10 4 4 10 = 1600 C