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

Binomial theorem

Grade 12 Math

The Binomial Theorem is another method to help us expand binomials in a faster manner. It is particularly useful when we work on binomial expansions that involve binomials raised to high powers.

12th Grade12th