Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
The binomial theorem
Expand integer powers of binomials using combinatorial coefficients
Problem types
- A coefficient in a binomial expansion
Builds on
- Adding, subtracting, and multiplying polynomials
- Permutations and combinations · Useful idea
The coefficient of the term with k factors of a in (a + b)ⁿ counts the ways to choose which k of the n factors give a. That count is the combination C(n, k).
- Proving a polynomial identity · Earlier idea
The identities for (a + b)² and (a + b)³ are the cases n = 2 and n = 3 of the binomial expansion. The binomial theorem gives the coefficients for every n.
Leads to
- The multiplication rule for probability · See it applied
The probability of exactly k successes in n independent trials multiplies the success probabilities and a binomial coefficient.
- A distribution from a sample space · See it applied
The probability of k successes in n independent trials with success chance p is C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ. These are the terms of the expansion of (p + (1 − p))ⁿ, which sum to 1.
- Using expected value to compare choices · See it applied
The expected number of successes in n independent trials is n times the success probability, the mean of the binomial distribution.
Related skills
- Sums of a finite geometric sequence · Related idea
Sigma notation writes a sum of terms, and the binomial theorem is a sum whose terms use binomial coefficients.
- Factoring over the complex numbers · Shared foundations
- Dividing polynomials · Shared foundations
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