Gain insights into dynamic programming, explore recursion basics, and delve into advanced techniques like Bottom-Up optimization. Discover ways to solve complex problems more efficiently with hands-on coding challenges.

Dynamic programming is something every developer should have in their toolkit. It allows you to optimize your algorithm with respect to time and space — a very important concept in real-world applications.

In this course, you’ll start by learning the basics of recursion and work your way to more advanced DP concepts like Bottom-Up optimization. Throughout this course, you will learn various types of DP techniques for solving even the most complex problems. Each section is complete with coding challenges of varying difficulty so you can practice a wide range of problems.

By the time you’ve completed this course, you will be able to utilize dynamic programming in your own projects.

Dynamic Programming in Python: Optimizing Programs for Efficiency

# Problem statement

Given a list of currency bills, you are required to count the number of ways in which you can represent a certain amount. For example, if you have only two kinds of bills, *$10* and *$20*, and you want to represent *$30*, there are only two ways:
 
- 3 bills of *$10* 
- 1 bill of *$20* and 1 bill of *$10*.

# Input

You will get a list of currency `bills` and the `amount` you are required to make as input. You have to count the number of ways the `amount` can be made using the currency bills provided in the `bills` list.


```
bills = [10, 20]
amount = 30
```

# Output

Your program should return the count of the number of ways `amount` can be represented using dollar bills given to you in the list `bills`.

```
countways([10, 20], 30) = 2
```

# Coding challenge

As you design the algorithm take special care that you do not overcount. $30 can be represented with $20+$10 as well as $10+$20, but these are the same thing. Therefore, you should not count both cases.


A simple recursive solution will timeout for large inputs, thus, you should try to write a dynamic programming algorithm. Start with a recursive solution and build up to a dynamic solution. Your solution may use a top-down or bottom-up approach. Set `stressTesting` to `False` to test your simple recursive solution.



In this lesson you will write an algorithm to solve a famous dynamic programming problem, finding the number of ways to represent an amount using bills and coins.

Chapter 1: From Recursion to Dynamic Programming

Chapter 2: Top-Down Dynamic Programming with Memoization

Chapter 3: Bottom-Up Dynamic Programming with Tabulation

Chapter 4: Practice Problems

Challenge: Number of Ways to Represent N Dollars

Problem statement