Binomial coefficients

High School

Definition

Numbers that The coefficients in the binomial equation. The coefficients must be a positive integer and when arranged in rows

Worked examples

\((x + y)^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3\)

The coefficients 1, 3, 3, 1 are the binomial coefficients for the third power.

\(\binom{4}{2} = 6\)

This notation reads "4 choose 2" and gives the binomial coefficient in row 4, position 2 of Pascal's triangle.

\((a + b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4\)

Row 4 of Pascal's triangle gives the coefficients: 1, 4, 6, 4, 1.

Common mistakes

\((x + y)^3 = x^3 + y^3\) → \((x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\) You must include all middle terms with their binomial coefficients, not just the first and last.
\(\binom{5}{2} = 5 \times 2 = 10\) → \(\binom{5}{2} = \frac{5!}{2!3!} = 10\) Use the combination formula, not simple multiplication.
The coefficients in row n add up to n → The coefficients in row n add up to \(2^n\) For example, row 3 has coefficients 1, 3, 3, 1 which sum to 8, not 3.

Where you'll use it next

You'll use binomial coefficients in probability and combinatorics to count combinations, in the Binomial Theorem to expand powers, and later in calculus for series expansions and Taylor polynomials.

Found in 1 StudyPug lesson

The Binomial Theorem

Grade 12 Math

Expand (x+y) to any power using the coefficients from row n of Pascal's triangle.

12th Grade12th