Math · College algebra · Concept
The binomial theorem and Pascal’s triangle
The binomial theorem expands (a + b)ⁿ into n + 1 terms: the term with bᵏ is C(n, k)aⁿ⁻ᵏbᵏ. The coefficients C(n, k) are row n of Pascal’s triangle, and they count the ways to choose b from k of the n factors.
Choose one term from every factor
(a + b)ⁿ is n factors of (a + b) multiplied together. Each term of the product picks a or b from every factor, so its powers of a and b always add to n. Collecting equal terms gives the coefficients.
The binomial theorem
The coefficient of aⁿ⁻ᵏbᵏ counts the ways to choose which k of the n factors supply b.
Counting the choices
The chosen factors form a set, so their order does not matter. There are n!/(k!(n − k)!) such sets, the binomial coefficient “n choose k”.
Pascal’s triangle
Each entry is the sum of the two entries above it, and row n lists C(n, 0) to C(n, n). The rule works because a choice of k factors either includes the last factor or does not.
| n | Row n |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
Signs and coefficients inside the terms
Substitute each whole term, sign and coefficient included. In (x − 2)ⁿ, b = −2, so the terms alternate in sign; in (2x + 1)ⁿ, a = 2x, and the 2 is raised to each power along with x.
One term without the whole expansion
The term containing bᵏ is C(n, k)aⁿ⁻ᵏbᵏ. Match the power you want, find k, and compute just that term.
Common mistakes
- Dropping the middle terms: (a + b)² is a² + 2ab + b², not a² + b².
- Losing the sign: in (x − 2)³, b = −2, so the terms alternate.
- Forgetting to raise a coefficient to its power: (2x)³ is 8x³, not 2x³.
- Using the wrong row: the power 3 uses row 3, which is 1, 3, 3, 1.
Key terms
- Binomial theorem
- The expansion (a + b)ⁿ = Σ C(n, k)aⁿ⁻ᵏbᵏ for k = 0 to n: each term chooses b from k of the n factors, in C(n, k) ways.
- Pascal’s triangle
- A triangular array in which each entry is the sum of the two above it. Row n lists the binomial coefficients C(n, 0) to C(n, n).
- Combination count
- The number of ways to choose k objects from n when order doesn’t matter: C(n, k) = n!/(k!(n − k)!). For example, 2 people can be chosen from 4 in 6 ways.
- Factorial
- For a positive integer n, n! is the product of the integers from 1 through n; 0! is defined as 1. Factorials count arrangements of distinct objects, and the calculator limits factorial input to integers from 0 through 170.
- Exponent
- The raised number in a power, telling how many times to multiply the base: 2³ = 2·2·2. A negative exponent means a reciprocal (2⁻³ = 1/8), and a fractional one means a root.
Work through an example
Expand (x − 2)³.
Expand (x − 2)³ with the binomial theorem →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Explore Pascal’s Triangle Check the expansion in Math Open worked example on a board The binomial theorem in Math ReferenceYour existing work stays on this device. Examples open as editable copies.