17.5.2. Application of Pigeonhole Principle

Try these first

1. 1.

   Provide an example illustrating the Pigeonhole Principle using fruits and containers.

   Hint

   Think of how many items you have versus containers.

2. 2.

   What is the midpoint of points (2, 6) and (8, 10)?

   Hint

   Use the midpoint formula.

3. 3.

   What is the fundamental idea behind the Pigeonhole Principle?

   • Every item must go in a different container
   • If there are more items than containers
   • at least one container must hold multiple items
   • Items must be distributed evenly

   Hint

   Consider a simple example of baskets and fruits.

4. 4.

   True or False: The midpoint of two points with non-integer coordinates is always non-integer.

   • True
   • False

   Hint

   Think about coordinate combinations.

5. 5.

   Identify 3 different sets of integers from 1 to 20 that, if choosing 8 at random, must yield pairs summing to 12. Show your reasoning.

   Hint

   Count the pairs available and what happens with extra selections.

6. 6.

   Using the Pigeonhole Principle, prove that for any selected set of 100 integers chosen from 1 to 200, at least two numbers will yield the same modulo 100.

   Hint

   Think about the range of results possible and total selections.