Binomial Theorem Cheatsheet

Cheat Codes & Shortcuts

• Binomial Theorem: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \), where \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• Binomial Coefficient: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), counts combinations.
• Pascal’s Triangle: Row \( n \) gives coefficients for \( (a + b)^n \).
• General Term: In \( (a + b)^n \), the \( (k+1) \)-th term is \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \).
• Sum of Coefficients: For \( (1 + 1)^n \), sum is \( 2^n \).
• Binomial Expansion for Non-Integer \( n \): \( (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \cdots \), valid for \( |x| < 1 \).
• Number of Terms: For \( (a + b)^n \), there are \( n+1 \) terms.
• Middle Term: For even \( n \), middle term at \( k = \frac{n}{2} \); for odd \( n \), two middle terms at \( k = \frac{n-1}{2}, \frac{n+1}{2} \).
• Application: Approximate \( (1 + x)^n \) for small \( x \).
• Combinatorial Identity: \( \sum_{k=0}^n \binom{n}{k} = 2^n \).

Quick Reference Table

Advice

Binomial Theorem Quick Tips

• Binomial Formula: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \).
• Pascal’s Triangle: Quick way to find coefficients for small \( n \).
• Middle Term: For even \( n \), use \( k = \frac{n}{2} \); for odd \( n \), check both middle terms.
• Non-Integer \( n \): Use \( (1 + x)^n \approx 1 + nx \) for small \( x \).
• Sum of Coefficients: Set \( a = b = 1 \) to get \( 2^n \).