Combinatorics, Double Counting

Double Counting is a combinatorial proof technique where a quantity is counted in two different ways. Setting the two resulting expressions equal to each other can lead to proving identities or inequalities. Questions require identifying a set that can be counted in multiple ways.

• Question

  By how much is the sum of all even numbers not exceeding `100` greater than the sum of all odd numbers not exceeding `100`?

  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• A TENNIS TOURNAMENT

  Four married couples played a "mixed double" tennis tournament, a man and a lady always playing against a man and a lady. But no person ever played with or against any other person more than once. Can you show how they all could have played together in the two courts on three successive days? This is a little puzzle of a quite practical kind, and it is just perplexing enough to be interesting.
  Topics:

  Combinatorics -> Pigeonhole Principle Combinatorics -> Double Counting
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 266