Offered A Liberal Arts Degree, Offered A Computer Engineer

242 Offered A Liberal Arts Degree116 Offered A Computer Engineering

242 offered a liberal arts degree. 116 offered a computer engineering degree. 467 offered a nursing degree. 35 offered a liberal arts degree and a computer engineering degree. 197 offered a liberal arts degree and a nursing degree. 95 offered a computer engineering degree and a nursing degree. 29 offered a liberal arts degree, a computer engineering degree, and a nursing degree. 36 offered none of these degrees. How many four-year colleges and universities were surveyed?

Paper For Above instruction

The problem involves calculating the total number of colleges and universities surveyed based on the provided data about the degrees they offer. The data indicates the number of colleges offering specific degrees, their overlaps, and those offering no degrees. To solve this, we employ the principle of inclusion-exclusion, a fundamental concept in set theory used to avoid double-counting when dealing with overlapping groups.

First, define the sets as follows: let L be the set of colleges offering a liberal arts degree, C the set offering a computer engineering degree, and N the set offering a nursing degree. Based on the data:

  • |L| = 242
  • |C| = 116
  • |N| = 467
  • |L ∩ C| = 35
  • |L ∩ N| = 197
  • |C ∩ N| = 95
  • |L ∩ C ∩ N| = 29
  • Colleges offering no degrees = 36

The total number of colleges can be calculated by summing all colleges offering at least one of these degrees and then adding the colleges offering none. The inclusion-exclusion principle states:

Total with at least one degree = |L| + |C| + |N| - |L ∩ C| - |L ∩ N| - |C ∩ N| + |L ∩ C ∩ N|

Substituting the known values:

Total with at least one degree = 242 + 116 + 467 - 35 - 197 - 95 + 29 = (825) - (327) + 29 = 527

Now, the total number of colleges surveyed, including those offering no degrees, is:

Total colleges = total with at least one degree + colleges offering none = 527 + 36 = 563

Therefore, the total number of colleges and universities surveyed is 563. This calculation accounts for overlapping degree offerings and ensures no double-counting, providing an accurate count of all institutions surveyed.

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