The Binomial Theorem provides a formula for expanding powers of binomials.
Overview
graph TD
A[Binomial Theorem] --> B[Expansion]
A --> C[General Term]
A --> D[Applications]
B --> B1[Pascal's Triangle]
C --> C1[Middle Term]
C --> C2[Greatest Term]Binomial Theorem
For positive integer n:
$$\boxed{(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r}$$ $$= \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + ... + \binom{n}{n}b^n$$Binomial Coefficient
$$\binom{n}{r} = {}^nC_r = \frac{n!}{r!(n-r)!}$$Properties
- $\binom{n}{0} = \binom{n}{n} = 1$
- $\binom{n}{r} = \binom{n}{n-r}$
- $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$
General Term
The $(r+1)^{th}$ term in $(a+b)^n$:
$$\boxed{T_{r+1} = \binom{n}{r} a^{n-r} b^r}$$Middle Term
If n is even: One middle term = $T_{(n/2)+1}$
If n is odd: Two middle terms = $T_{(n+1)/2}$ and $T_{(n+3)/2}$
Special Cases
$(1+x)^n$
$$= 1 + nx + \frac{n(n-1)}{2!}x^2 + ... + x^n$$$(1-x)^n$
$$= 1 - nx + \frac{n(n-1)}{2!}x^2 - ... + (-1)^n x^n$$Sum of Coefficients
Put $x = 1$: Sum of coefficients = $2^n$
Put $x = -1$: Sum of odd - even coefficients = $0$
Greatest Term
For $(1+x)^n$ where $x > 0$:
If $\frac{(n+1)x}{1+x}$ is an integer $= m$, then $T_m$ and $T_{m+1}$ are greatest.
Otherwise, $T_{[m]+1}$ is greatest (where [m] = floor).
Important Results
Sum: $C_0 + C_1 + C_2 + ... + C_n = 2^n$
Alternating: $C_0 - C_1 + C_2 - ... = 0$
$C_0 + C_2 + C_4 + ... = C_1 + C_3 + ... = 2^{n-1}$
$C_1 + 2C_2 + 3C_3 + ... + nC_n = n \cdot 2^{n-1}$
Multinomial Theorem
$$(x_1 + x_2 + ... + x_k)^n = \sum \frac{n!}{n_1! n_2! ... n_k!} x_1^{n_1} x_2^{n_2} ... x_k^{n_k}$$where $n_1 + n_2 + ... + n_k = n$
Practice Problems
Find the coefficient of $x^5$ in $(1+x)^{10}$.
Find the middle term of $(2x - \frac{1}{x})^{10}$.
Find the term independent of $x$ in $(x + \frac{1}{x^2})^{12}$.
Prove: $C_0^2 + C_1^2 + C_2^2 + ... + C_n^2 = \binom{2n}{n}$
Further Reading
- Permutations and Combinations - Counting principles
- Sequences and Series - Series summation