18.2.1. Age Group and Pigeonhole Principle Application

Try these first

1. 1.

   Define a strictly increasing subsequence.

   Hint

   Think about an example where numbers go up.

2. 2.

   What does the pigeonhole principle state?

   Hint

   Consider how this might apply to people and birthdays.

3. 3.

   What is a strictly increasing subsequence?

   • A sequence where elements decrease
   • A sequence where each element is equal
   • A sequence where each element is larger than the previous

   Hint

   Think of a climbing path.

4. 4.

   Does the pigeonhole principle allow for items to fit without overlap?

   • True
   • False

   Hint

   Consider how many people can fit into a car.

5. 5.

   Provide a proof using the pigeonhole principle to show that among any 10 individuals selected randomly, there must be at least two who share the same birthday month.

   Hint

   Think of how many months the year has versus the number of persons.

6. 6.

   Given a sequence of 10 distinct integers, demonstrate using theorems related to subsequences that there must be increasing subsequences of at least 4 numbers.

   Hint

   Think about how they can be grouped and compared in ranges.