Quality control in a factory pulls 40 parts with paint packa

Quality control in a factory pulls 40 parts with paint, packaging, or electronics defects from an assembly line. Of these, 28 had a paint defect, 17 had a packaging defect, 13 had an electronics defect, 6 had both paint and packaging defects, 7 had both packaging and electronics defects, and 10 had both paint and electronics defects. Did any part have all three types of defect?

Solution

a)

Yes there are the parts which have all three type of defects
First take A = parts with paint defects
Take K = parts with packaging defects
Take E = parts with electronics defects

| A K E |   = 40
| A |   = 28
| K |   = 17
| E |   = 13
| A K |   = 6
| K E |   = 7
| A E |   = 10

By taking the three-set version of the Principle of Inclusion and Exclusion we know that:

1.| A K E | = |A| + |K| + |E| – | A K | – | K E | – | A E | + | A K E |
2.40 = 28 + 17 + 13 – 6 – 7 – 10 + | A K E |
3.40 – 28 – 17 – 13 + 6 + 7 + 10 = | A K E |
4. 5 = | A K E |