# Lecture 3 – Binomial Coefficients, Lattice Paths, & Recurrences

You may want to download the lecture slides that were used for these videos (PDF).

## 1. Binomial Coefficients & Distributing Objects

Here, we relate the binomial coefficients to the number of ways of distributing m identical objects into n distinct cells. (3:51)

## 2. Distributing Objects with Different Restrictions

We see how the number of ways of distributing objects changes as the restrictions on how we distribute the objects change.  For example, we may not require that each cell receives an object.  (9:01)

## 3. More on Distributing Objects: Good = All – Bad

Depending on the constraints in the problem, it is often easier to compute the number of ways of distributing the objects (the “good” distributions) by counting the number of distributions without constraints and subtracting the ones we aren’t interested in (the “bad” distributions). (7:29)

## 4. Lattice Paths

If we can only move up and right, how many different ways can we make a path on a grid between two points? (7:39)

## 5. Catalan Numbers: Lattice Paths Not Above the Diagonal

If we can’t move above the diagonal, and again can only move up and right, how many different ways can we make a path on a grid between two points? (16:29)

(Note: Video 6 was a duplicate and has been removed.)

## 7. Parentheses & Catalan Numbers

The Catalan Numbers are ubiquitous.  For example, they count the number of ways of parenthesizing an algebraic expression. (2:29)

## 8. Deriving Recurrence Relations

In this video we explore how to derive recurrence relations for five different problems. (22:14)

## 9. Are Recurrence Equations Good Enough?  (Plus, a Comment on Division.)

Once we have a recurrence relation, do we want to know more about the sequence of numbers that it generates? (2:46)