Level 7: The binomial theorem

Counting and combinations · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In level 5 the numbers \(\dbinom{n}{k}\) counted teams and filled Pascal's triangle. In this level they turn up somewhere unexpected: in algebra, as the numbers you get when you multiply out \((a + b)^n\). That one link, the binomial theorem, turns counting facts into algebra facts and algebra facts back into counting facts. Along the way you meet your first real proofs by counting, where an identity is proved by counting one set in two different ways.

Lesson	You will be able to
Expanding by choosing	Multiply out any product of brackets by choosing one term from each, and explain why the coefficients of \((a + b)^n\) are choose numbers
The binomial theorem	State the theorem with \(\Sigma\) notation, prove it, and find any single term of an expansion such as the \(x^5\) term of \((2x - 3)^8\)
Identities from the theorem	Get row sums, alternating sums and other identities by putting numbers in for \(a\) and \(b\), and explain each one by counting too
Patterns in Pascal's triangle	Prove the hockey-stick identity, find the triangular and Fibonacci numbers in the triangle, and explain the Sierpinski picture of its odd entries
Proof by counting	Prove identities such as \(k\dbinom{n}{k} = n\dbinom{n-1}{k-1}\) and Vandermonde's identity by counting one set in two ways
Multinomial coefficients	Count arrangements of words like AABBBC, expand \((a + b + c)^n\), and find any coefficient as \(\dfrac{n!}{p!\,q!\,r!}\)

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