Original practice · fully worked
Original variant: junctions flagged by overlapping detectors
A utility inspects its junctions with thermal, acoustic, and vibration detectors. The detectors flag 140, 120, and 100 junctions, respectively. The three pairwise intersection counts are 40, 30, and 20, and 10 junctions are flagged by all three detectors. Calculate the number of junctions flagged by at least one detector.
- A
260 - B
270 - C
280 - D
290 - E
360
Answer and calculation
Inclusion–exclusion adds the three detector counts, subtracts the three pairwise intersections, and restores the all-three intersection once. This gives 280 junctions and choice C.
Setup
Setup
Let T, A, and V denote the three detector flag sets.
∣T∣+∣A∣+∣V∣=140+120+100=360 Model
Model
Apply inclusion–exclusion to the union of the three flag sets.
∣T∪A∪V∣=∣T∣+∣A∣+∣V∣ −∣T∩A∣−∣T∩V∣−∣A∩V∣+∣T∩A∩V∣ Compute
Compute
Substitute the three pairwise counts and the all-three count.
∣T∪A∪V∣=360−(40+30+20)+10 Answer
Answer
Exactly 280 junctions are flagged by at least one detector.
280(C) 