🎯 Binomial Theorem & Series

Master binomial expansions, coefficient calculations, and estimation techniques. Essential for AMC 12 and appears in AMC 10.

🎯 Key Ideas

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

Binomial Coefficients: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ count combinations

Applications: Probability, approximation, coefficient problems

📊 Binomial Theorem

Statement

For any real numbers $a, b$ and non-negative integer $n$: $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$

Special Cases

  • $(1+x)^n$: $(1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k$
  • $(x+y)^n$: $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$

Example: Expand $(2x-3)^4$

Solution:

  • $(2x-3)^4 = \sum_{k=0}^4 \binom{4}{k} (2x)^{4-k} (-3)^k$
  • $= \binom{4}{0}(2x)^4 + \binom{4}{1}(2x)^3(-3) + \binom{4}{2}(2x)^2(-3)^2 + \binom{4}{3}(2x)(-3)^3 + \binom{4}{4}(-3)^4$
  • $= 1 \cdot 16x^4 + 4 \cdot 8x^3(-3) + 6 \cdot 4x^2(9) + 4 \cdot 2x(-27) + 1 \cdot 81$
  • $= 16x^4 - 96x^3 + 216x^2 - 216x + 81$

🔢 Binomial Coefficients

Definition

$$\binom{n}{k} = \frac{n!}{k!(n-k)!} = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$$

Properties

  • Symmetry: $\binom{n}{k} = \binom{n}{n-k}$
  • Pascal’s Identity: $\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$
  • Sum: $\sum_{k=0}^n \binom{n}{k} = 2^n$

Example: Calculate $\binom{7}{3}$

Solution:

  • $\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7!}{3!4!} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = \frac{210}{6} = 35$

📈 Pascal’s Triangle

Construction

  • Row $n$ contains coefficients for $(a+b)^n$
  • Each entry is sum of two entries above it
  • First and last entries in each row are 1

First Few Rows

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

Example: Use Pascal’s triangle to expand $(x+y)^3$

Solution:

  • Row 3: $1, 3, 3, 1$
  • $(x+y)^3 = 1x^3 + 3x^2y + 3xy^2 + 1y^3 = x^3 + 3x^2y + 3xy^2 + y^3$

🎯 Coefficient Problems

Finding Specific Coefficients

To find coefficient of $x^k$ in $(ax+b)^n$:

  1. General term: $\binom{n}{r} (ax)^{n-r} b^r$
  2. Exponent of $x$: $n-r = k$, so $r = n-k$
  3. Coefficient: $\binom{n}{n-k} a^{n-(n-k)} b^{n-k} = \binom{n}{k} a^k b^{n-k}$

Example: Find coefficient of $x^3$ in $(2x+1)^5$

Solution:

  • General term: $\binom{5}{r} (2x)^{5-r} (1)^r$
  • Exponent: $5-r = 3$, so $r = 2$
  • Coefficient: $\binom{5}{2} \cdot 2^2 \cdot 1^3 = 10 \cdot 4 \cdot 1 = 40$

📊 Sums of Binomial Coefficients

Common Sums

  • All coefficients: $\sum_{k=0}^n \binom{n}{k} = 2^n$
  • Even coefficients: $\sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} = 2^{n-1}$ (for $n \geq 1$)
  • Odd coefficients: $\sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \binom{n}{2k+1} = 2^{n-1}$ (for $n \geq 1$)

Example: Find $\sum_{k=0}^5 \binom{5}{k}$

Solution:

  • $\sum_{k=0}^5 \binom{5}{k} = 2^5 = 32$

🔄 Binomial Series (Infinite)

For $|x| < 1$:

$$(1+x)^\alpha = \sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$$

where $\binom{\alpha}{k} = \frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k!}$ for any real $\alpha$.

Special Cases

  • $\alpha = -1$: $\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots$
  • $\alpha = \frac{1}{2}$: $\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \cdots$

🎯 AMC-Style Worked Example

Problem: Find the coefficient of $x^4$ in the expansion of $(1+2x+3x^2)^6$.

Solution:

  1. Rewrite: $(1+2x+3x^2)^6 = (1+x(2+3x))^6$
  2. Apply binomial theorem: $\sum_{k=0}^6 \binom{6}{k} 1^{6-k} (x(2+3x))^k$
  3. Focus on $x^4$ terms: Need $k$ such that $(x(2+3x))^k$ contributes $x^4$
  4. Analyze each $k$:
    • $k=0$: No $x$ terms
    • $k=1$: $x(2+3x) = 2x + 3x^2$ → no $x^4$
    • $k=2$: $(x(2+3x))^2 = x^2(2+3x)^2 = x^2(4+12x+9x^2) = 4x^2+12x^3+9x^4$ → coefficient $9$
    • $k=3$: $(x(2+3x))^3 = x^3(2+3x)^3$ → highest power is $x^6$ → no $x^4$
    • $k \geq 4$: Higher powers → no $x^4$
  5. Calculate: For $k=2$, coefficient is $\binom{6}{2} \cdot 9 = 15 \cdot 9 = 135$

Answer: 135

🔍 Common Traps & Fixes

Trap: Wrong exponent calculation

Fix: In $(ax+b)^n$, the exponent of $x$ in term $\binom{n}{r} (ax)^{n-r} b^r$ is $n-r$.

Trap: Forgetting to multiply by coefficient

Fix: The coefficient includes both the binomial coefficient and the powers of $a$ and $b$.

Trap: Confusing finite and infinite series

Fix: Binomial theorem is for finite $n$; binomial series is for infinite expansion with $|x| < 1$.

Trap: Wrong range of convergence

Fix: Infinite binomial series converges only when $|x| < 1$.

📋 Quick Reference

Binomial Theorem

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

Binomial Coefficients

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

Common Sums

  • $\sum_{k=0}^n \binom{n}{k} = 2^n$
  • $\sum_{k=0}^n \binom{n}{k} x^k = (1+x)^n$

Pascal’s Triangle Properties

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

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