In a business school, only three specializations are offered: marketing, finance and operations. A student can opt for either one, or two, or no specializations. In the current batch of 120 students, 60 are specializing in marketing, 50 are specializing in finance and 30 are specializing in operations. During the placement process, all students specializing either in marketing or in finance are shortlisted for consulting job interviews.
If 45 students are not shortlisted for consulting job interviews, what is the MINIMUM possible number of students specializing in both marketing and finance?
A
Step-by-Step Solution
Key idea: This is a set theory problem with exact unions, recognizable by the total population, the sizes of individual sets, and the exact number of elements outside the union.
Step 1: Total students = 120.
Step 2: Students not shortlisted = 45.
Step 3: The problem states "all students specializing either in marketing or in finance are shortlisted". This means the set of shortlisted students is exactly the union of Marketing (M) and Finance (F).
Step 4: Number of shortlisted students = .
Step 5: Therefore, .
Step 6: We are given and .
Step 7: Use the inclusion-exclusion principle for two sets: .
Step 8: Substitute the known values: .
Step 9: .
Step 10: Since all the values in the equation are fixed constants, the number of students specializing in both is exactly 35. The minimum possible number is therefore 35.
Answer: 35