In a village of 100 households, if 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player, what is the maximum number of households that could have all three devices?

To find the maximum number of households that could have all three devices—DVD players, cell phones, and MP3 players—we can use the principle of inclusion-exclusion along with some logical reasoning.

We know the following data points from the problem:

• 75 households have at least one DVD player.

• 80 households have at least one cell phone.

• 55 households have at least one MP3 player.

• The total number of households is 100.

To determine the maximum possible number of households that own all three devices, we can start by analyzing how many households do not own each type of device:

• Households without a DVD player: 100 - 75 = 25

• Households without a cell phone: 100 - 80 = 20

• Households without an MP3 player: 100 - 55 = 45

Next, we need to ensure that we maximize the overlap (those who have all three devices).

To find this, we can use the formula for the union of sets. The maximum number of households that can have all three devices occurs when the overlaps are as large as possible. We need to account for the total number of devices owned by the households, which cannot exceed the