The Binomial Theorem

By the end of this lesson, you will be able to expand binomial expressions using the Binomial Theorem, find specific terms in a binomial expansion, and compute binomial coefficients using Pascal's Triangle and the combination formula.

The Big Idea

Expanding $(a + b)^n$ by repeated FOIL gets unwieldy past $n = 3$. The Binomial Theorem gives a direct formula using combinations. The coefficients, arranged in Pascal's Triangle, appear throughout probability and combinatorics — this lesson bridges algebra and counting.

Pascal's Triangle

n=0:    1
n=1:   1 1
n=2:  1 2 1
n=3: 1 3 3 1
n=4: 1 4 6 4 1
n=5: 1 5 10 10 5 1

Each row gives the coefficients of $(a+b)^n$:

Each interior entry = sum of the two entries above it.

Binomial Theorem

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

where $\dbinom{n}{k} = \dfrac{n!}{k!(n-k)!}$ is the binomial coefficient ($n$ choose $k$).

The $(k+1)$th term is $\dbinom{n}{k}a^{n-k}b^k$.

Example 1 — Full Expansion

Expand $(x + 2)^4$.

$= \binom{4}{0}x^4 + \binom{4}{1}x^3(2) + \binom{4}{2}x^2(4) + \binom{4}{3}x(8) + \binom{4}{4}(16)$

$= x^4 + 4(2)x^3 + 6(4)x^2 + 4(8)x + 16$

$= x^4 + 8x^3 + 24x^2 + 32x + 16$

Example 2 — Negative Term

Expand $(2a - 3b)^3$.

$= (2a)^3 - 3(2a)^2(3b) + 3(2a)(3b)^2 - (3b)^3$

$= 8a^3 - 3(4a^2)(3b) + 3(2a)(9b^2) - 27b^3$

$= 8a^3 - 36a^2b + 54ab^2 - 27b^3$

Finding a Specific Term

The $(k+1)$th term of $(a+b)^n$ is $\binom{n}{k}a^{n-k}b^k$.

Example: Find the 4th term of $(x - 2)^7$.

$k = 3$ (4th term is $k+1 = 4$, so $k = 3$):

$\binom{7}{3}(x)^4(-2)^3 = 35 x^4 (-8) = -280x^4$

Binomial Coefficients

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Examples:

$\binom{6}{2} = \frac{720}{2 \times 24} = 15$

$\binom{8}{5} = \binom{8}{3} = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{6} = 56$

Symmetry: $\binom{n}{k} = \binom{n}{n-k}$

Connection to Probability

Binomial coefficients count the number of ways to choose $k$ items from $n$ without replacement (combinations). In probability, the binomial distribution uses $\binom{n}{k}p^k(1-p)^{n-k}$ — the Binomial Theorem with $a = p$ and $b = 1-p$.

Common Mistakes

Mistake 1: Wrong term index ($k$ starts at 0)

The first term ($k=0$) is $\binom{n}{0}a^n b^0 = a^n$. The 4th term uses $k=3$.