The Binomial Theorem

Binomial Theorem Formula

For any positive integer $n$ :

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

Expanded:

$(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n$

Binomial Coefficients

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

Properties

$\binom{n}{0} = \binom{n}{n} = 1$

$\binom{n}{1} = \binom{n}{n-1} = n$

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

$\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$

Pascal's Triangle

Each number is the sum of the two numbers above it:

Row 0:           1
Row 1:          1 1
Row 2:         1 2 1
Row 3:        1 3 3 1
Row 4:       1 4 6 4 1
Row 5:      1 5 10 10 5 1
Row 6:     1 6 15 20 15 6 1

Row $n$ gives the coefficients for $(a + b)^n$

Common Expansions

$(a + b)^2$

$(a + b)^2 = a^2 + 2ab + b^2$

$(a - b)^2$

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

$(a + b)^3$

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

$(a - b)^3$

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

$(a + b)^4$

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

$(a + b)^5$

$(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$

Finding a Specific Term

The $(k+1)$ th term in $(a + b)^n$ :

$T_{k+1} = \binom{n}{k} a^{n-k} b^k$

Example

Find the 4th term of $(2x + 3)^5$ :

$k = 3$ (4th term means $k+1 = 4$ )
$T_4 = \binom{5}{3}(2x)^{5-3}(3)^3$
$T_4 = 10 \cdot 4x^2 \cdot 27 = 1080x^2$

Binomial Series (for any real n)

When $|x| < 1$ :

$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$

Special Cases

Sum of Coefficients

$(1 + 1)^n = 2^n = \text{sum of all binomial coefficients in row } n$

Alternating Sum

$(1 - 1)^n = 0 = \binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \binom{n}{3} + \cdots$

Middle Term

For $(a + b)^n$ :

If $n$ is even: One middle term at position $(n/2 + 1)$
If $n$ is odd: Two middle terms at positions $(n+1)/2$ and $(n+3)/2$

Applications

Probability: $\binom{n}{k}$ gives number of ways to choose $k$ items from $n$
Approximations: $(1 + x)^n \approx 1 + nx$ for small $x$
Combinatorics: Counting problems
Polynomial expansion: Simplifying expressions