Chapter 7

Binomial series and proof by induction

Chapter description
In this chapter we find out how to do binomial expansions, and see how they can describe some real-life situations. We also look at a new method of proving mathematical statements.
The chapter is divided into the following sections.

  1. Binomial series for positive whole numbers
    1. Looking for the patterns
    2. Permutations or arrangements
    3. Combinations or selections
    4. How selections give binomial expansions
    5. Writing down rules for binomial expansions
    6. Linking Pascal's triangle to selections
    7. Some more binomial examples

  2. Some applications of binomial series and selections
    1. Tossing coins and throwing dice
    2. What do the probabilities we have found mean?
    3. When is a game fair? (Or are you fair game?)
    4. Lotteries: winning the jackpot ... or not

  3. Binomial expansions when n is not a positive whole number
    1. Can we do it? If so, when?
    2. Working out some expansions
    3. Dealing with slightly different situations

  4. Mathematical induction
    1. Truth from patterns ... or false mirages?
    2. Proving the binomial theorem by induction
    3. Two non-series applications of induction
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