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Introduction
In Theorem 1.1.4 and Section 3.2, we saw that the number of handshakes in a group of \(n\) people or the number of edges of the complete graph \(K_n\text{,}\) is
Similarly, in Section 2.2 and Section 6.4, we saw that the number of ways to order \(n\) distinct objects, or the number of permutations of the set \(\{1, 2, 3, \ldots,
n\}\text{,}\) is
In this chapter, we explore sums and products. We begin with two activities involving sums in Section 9.1, one of which introduces a representation of integers using sums of factorials. We also introduce summation and product notation. Section 9.2 continues our discussion of representing integers using sums, this time of binary, hexadecimal, and other base representations. Sums and products of an arbitrary number of integers are defined recursively, and so it is reasonable to investigate if there are explicit formulas for sums and products in Section 9.3. We return to counting and polynomial algebra in Section 9.4 where we explore recursive and explicit formulas for \(\binom{n}{k}\) as well as patterns in the Arithmetic triangle, some of which involve sums. Finally, in Section 9.5 we look at techniques for proving a summation or product formula including proof by mathematical induction, combinatorial proof, and other proofs.
