二项式数

Introduction: Choosing k From n

Definition of Factorial

What is $4!$?

Definition of Binomial Coefficients

What is $\binom{4}{2} = \frac{4!}{2!\cdot 2!}$?

Symmetry

By symmetry $\binom{7}{5} = \binom{7}{?}$, what is the question mark?

Stifel's Relation

By Stifel's relation $\binom{5}{2} = \binom{4}{2} + \binom{4}{?}$, what is the question mark?

Pascal's Triangle

Row 5 of Pascal's triangle is $1, 5, 10, 10, 5, 1$. What is $\binom{5}{2}$?

Example: The Column Theorem and Sum of Squares

Compute $\binom{6}{3} = \frac{6!}{3!\cdot 3!}$; enter an integer.

Use the column theorem to compute $\sum_{j=0}^{3}\binom{j}{0} = \binom{0}{0}+\binom{1}{0}+\binom{2}{0}+\binom{3}{0}$; enter an integer.

Row 4 of Pascal's triangle is $1, 4, 6, 4, 1$. Find $\binom{4}{3}$; enter an integer.

What are the definition, the symmetry, and Stifel's relation of binomial coefficients?

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$ involves division, so why is it necessarily an integer? Explain using Stifel's relation and induction.

Comprehensive Check

Lesson Summary