二项式公式

Introduction: Pascal's Triangle and Binomials

Newton's Binomial Formula

The $j$-th term of $(x+y)^n$ is $\binom{n}{j}x^{?}y^j$. What is the question mark (the power of $x$)?

Example: Proving the Binomial Formula by Induction

In the induction proof of the binomial formula, what is used to combine like terms in the induction step?

Special Case One: Sum of Binomial Coefficients

What is $\sum_{j=0}^{5}\binom{5}{j} = 2^5$?

Special Case Two: Alternating Sum

What is $\sum_{j=0}^{n}(-1)^j\binom{n}{j}$ (for $n\ge 1$)?

Compute $(1+1)^5 = \sum_{j=0}^{5}\binom{5}{j}$; enter an integer.

What is the coefficient of $x^3$ in $(1+x)^6$? Enter an integer.

What is the coefficient of the $x^2 y^2$ term in the expansion of $(x+y)^4$? Enter an integer.

Common Error: Power Pairing

The $j$-th term of $(x+y)^n$ is $\binom{n}{j}x^{n-j}y^j$. What is the sum of the powers of $x$ and $y$?

What are the binomial formula and its two special cases?

Use the binomial formula to explain $\sum_{j=0}^{n}\binom{n}{j} = 2^n$, and state its combinatorial interpretation.

Comprehensive Check

Lesson Summary