Equations · Algebra and geometry
Binomial theorem
How to expand a sum raised to a whole-number power, term by term.
(x + y)ⁿ = ∑ₖ₌₀ⁿ C(n, k) xⁿ⁻ᵏ yᵏ
Aethel says
the quantity x plus y, to the power of n, equals the sum from k equals 0 to n of n choose k, times x to the power of n minus k, times y to the power of k
What it says
Multiplying out x plus y, n times over, gives a sum of n plus 1 terms. Each term has x to some power and y to the remaining power, and its coefficient counts the ways to pick which factors contribute a y.
C(n, k), also written as n over k in tall brackets, is the number of ways to choose k things from n, equal to n factorial divided by k factorial times n minus k factorial. These coefficients are the rows of Pascal's triangle.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| n | n | the power, a whole number zero or more |
| ∑ₖ₌₀ⁿ | the sum from k equals 0 to n of | add up the terms for k = 0, 1, 2, … up to n |
| C(n, k) | n choose k | the binomial coefficient: the number of ways to choose k items from n |
Where you meet it
Algebra and combinatorics, and the binomial distribution in probability.
Questions
How do you say Binomial theorem out loud?
(x + y)ⁿ = ∑ₖ₌₀ⁿ C(n, k) xⁿ⁻ᵏ yᵏ is read: “the quantity x plus y, to the power of n, equals the sum from k equals 0 to n of n choose k, times x to the power of n minus k, times y to the power of k.” That is exactly what Aethel says when it meets the formula in a book.
What does n mean in Binomial theorem?
n (said “n”) is the power, a whole number zero or more.
What does ∑ₖ₌₀ⁿ mean in Binomial theorem?
∑ₖ₌₀ⁿ (said “the sum from k equals 0 to n of”) is add up the terms for k = 0, 1, 2, … up to n.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.