---
tags: mth225, dailyprep
---

# Daily Prep 4.3 -- MTH 225

## Overview

This lesson is part 2 of a look at the **binomial coefficient** which we explored and defined in the previous lesson. There's no new content to learn here, but we will be looking at two questions that have surprising connections to the binomial coefficient. You'll learn about those in the exercises.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

## Resources for learning

**Video:** **There's no new video to watch this time!** However, do review any of the five videos from the previous lesson if you need to go over the basics of the binomial coefficient.

**Text:** Also nothing to read here -- in fact **Stay Away from the Textbook** this time because there are spoilers in it! *However, please read the background material in the "Exercises" section below.*

## Exercises

There's some background you need to know first before you can do the exercises.

### Lattice Paths

Back in high school algebra you plotted points in the $xy$-plane. Those points could have any real numbers as coordinates. If we restrict our view to just *integers* as coordinates --- like $(2,-3)$ but excluding $(1/2, -3)$ and so on --- we get what is called an **integer lattice**. It's called a lattice because it's just a bunch of discrete dots:


If we lived on this lattice, and wanted to travel from one dot to another only by moving through other dots, the path we would trace out is called a **lattice path**. Specifically a *lattice path* from one lattice point to another is any *shortest possible* path between the two points that moves only horizontally or vertically. For example, here are three examples of lattice paths from $(0,0)$ to $(3,2)$:


But the path in red from $(0,0)$ to $(4,3)$ would *not* be a valid path because it is not as short as possible --- it wanders around too much.


In fact you might realize that in order to have a "shortest possible path" from $(0,0)$ to anywhere in the first quadrant, *you can only move up or to the right*. Movement down or to the left lengthens the path unnecessarily and causes you to have to backtrack.

It's a little like going from one building to another in the downtown of a city. You can't literally walk straight from the start to the destination; you have to follow the streets in the city block. In fact lattice paths are important for computer modeling of exactly that situation (traffic flow, etc.).

The question you're going to explore in the exercises is **How many possible lattice paths are there from $(0,0)$ to some other point $(a,b)$ in the first quadrant?** For example you can see from the picture above that there are at least three different lattice paths from $(0,0)$ to $(3,2)$. But how many are *possible*?

### Binomials and their expansion

A *binomial* is an algebra term that refers to the sum of two objects, like $x+2$ or $a+b$. In high school algebra, you learned how to **expand** a binomial using the FOIL method and repeated multiplication. For example:

$$(a+b)^2 = a^2 + 2ab + b^2$$

and

$$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$

The numbers that appear in front of the variables in an expression like this are called **coefficients**. For example, the coefficient on the $ab^2$ term in $(a+b)^3$, is $3$.

We are not about to waste time expanding huge powers of binomials, like $(a+b)^{100}$, by hand. What are are going to do instead is let computers handle the algebra and use our human brains to look for patterns that will describe what's happening. One of the best computer tools out there for doing symbolic algebra is **Wolfram|Alpha** (http://wolframalpha.com), a free service that does all kinds of symbolic math and other computations. If you go to Wolfram|Alpha and just enter in `expand (a+b)^6` for example, you'll get the expansion you asked for:


We want to focus on the coefficients of these expansions. For instance, as you can see in $(a+b)^6$, the coefficient on the $a^2b^4$ term is $15$; the coefficient on the $ab^5$ term is $6$; the coefficient on the $a^6$ term is $1$ (we don't actually write the $1$ however because it's multiplied).

If you're seeing some familiar faces in these coefficients, you're on the right track.

### Here are the actual exercises

As usual, go to this Google Form for the exercises:

https://docs.google.com/forms/d/e/1FAIpQLSdcW3AqA9HFvB95-Xmgjge-J9HkfGsl3Ic9Rn995qe7ayAivA/viewform

## Submission and grading

**Submitting your work:** Your work is submitted when you submit the Google Form. You should receive an email receipt indicating that the work was submitted successfully.

**How this is graded:** The pre-class portion of the Daily Prep is graded either 0 points or 1 point, on the basis of completeness and effort. Wrong answers are not penalized. Earning a "1" requires that you:

- Turn the work in before its deadline;
- Leave no item blank or skipped, even accidentally; and
- Give a good-faith effort at a correct answer on every non-optional item.

More information can be found in the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document.

When you arrive for the class meeting, you'll be put into a group of 2-3 to complete a quiz over this material, which will be graded on a 0/1 scale on the basis of correctness.