In this problem, we need to determine the minimum number of rooms required to conduct all nine seminars, given their overlapping timings. We can approach this problem using graph coloring. Each seminar represents a vertex in a graph, and an edge exists between two vertices if the seminars overlap and hence cannot be scheduled in the same room. Our goal is to color the graph (assign rooms) with the fewest number of colors (rooms) such that no two adjacent vertices share the same color. Let's analyze the overlaps based on the given information: Seminar P overlaps with V, X, and U. Seminar Q overlaps with R, W, V, and T. Seminar R overlaps with Q, W, V, and T. Seminar S overlaps with W and T. Seminar V overlaps with W. From these conditions, we can construct a graph where vertices represent seminars, and edges represent overlaps: P has edges to V, X, U Q has edges to R, W, V, T R has edges to Q, W, V, T S has edges to W, T V has edges to P, Q, R, W Let’s find the chromatic number of this graph, which is the minimum number of rooms needed: The maximum degree in the graph based on the list is 5, which is seminar R and V. According to graph theory, the chromatic number is at most one more than the maximum degree of the graph. You would typically have a chromatic number of 4 in such a well-connected scenario, as we see overlaps considering R’s and V’s full connectivity. A typical grouping to minimize rooms: - Room 1: P, Q, S (do not overlap directly) - Room 2: R, X (do not overlap directly) - Room 3: V (needs separate due to overlap with P, Q, R) - Room 4: W, T, U (do not overlap directly) Therefore, the minimum number of rooms required is (C) 4.

difficult to solve the set In a collegenine seminars are to be conducted. students are allowed to attend whichever seminar they want to. But timings of some seminars overlap with each other. It is known that1. The timing of seminar P overlaps with that of V, X and U.2. The timing seminar of Q overlaps with that of R, W, V and T3. The timing seminar R overlaps with that of Q, W, V and T4. The timing of seminar S overlaps with that of W and T5. The timing of seminar V overlaps with that of W.Q1. What is the minimum possible number of rooms required to conduct all the nine seminars?(A) 2(B) 3(C) 4(D) 5

difficult to solve the set In a collegenine seminars are to be conducted. students are allowed to attend whichever seminar they want to. But timings of some seminars overlap with each other. It is known that1. The timing of seminar P overlaps with that of V, X and U.2. The timing seminar of Q overlaps with that of R, W, V and T3. The timing seminar R overlaps with that of Q, W, V and T4. The timing of seminar S overlaps with that of W and T5. The timing of seminar V overlaps with that of W.Q1. What is the minimum possible number of rooms required to conduct all the nine seminars?(A) 2(B) 3(C) 4(D) 5