In a car repair shop there are 50 cars to be repaired . It is estimated that 23 of that cars need new brakes , and 34 of that cars need to repair the exhaust system .

a. How many cars are in need of repair at least two?
b . What most cars that are not in need of repair both?

By the pigeon hole principle (where each hole houses one car), it is impossible to have 23+34=57 repairs in 50 lifts (cars). So the surplus must go to cars that need both repairs.

This is the least number of cars that need both repairs. Assuming that all cars need at least one repair, then the rest of the cars need exactly one repair.

To find the number of cars that need repair in at least two areas, we need to determine the intersection of the cars that need new brakes and the cars that need to repair the exhaust system.

a. To find the number of cars that need repair in at least two areas, we can use the concept of set intersection. We calculate the intersection of the sets of cars that need new brakes and the sets of cars that need to repair the exhaust system.

Intersection = Number of cars that need new brakes ∩ Number of cars that need to repair the exhaust system

Therefore, the number of cars that need repair in at least two areas is the intersection of 23 cars that need new brakes and 34 cars that need to repair the exhaust system.
Intersection = 23 ∩ 34 = (number of cars in need of repair at least two)

b. To find the number of cars that don't need repair in both areas, we need to subtract the sum of the cars that need new brakes and the cars that need to repair the exhaust system from the total number of cars.

Number of cars that don't need repair in both = Total number of cars - (Number of cars that need new brakes + Number of cars that need to repair the exhaust system)

Therefore, the number of cars that don't need repair in both areas is calculated as:
Number of cars that don't need repair in both = 50 - (23 + 34)
(Number of cars that don't need repair in both) = (result of the calculation)

By performing the necessary calculations, you can find the specific numbers for these questions.