Consider the following variation on the change-making...

Key Concepts

Change-making Problem

This problem involves determining a combination of coin denominations that sum to a target value. It traditionally focuses on finding the minimum number of coins required; however, variations can impose additional constraints, such as using each denomination at most once. This constraint changes the nature of the problem by limiting the available supply for each coin type.

Subset Sum Problem

The subset sum problem asks whether a subset of given numbers can be selected such that their sum equals a target value. When each coin denomination can be used at most once, the change-making problem effectively becomes an instance of the subset sum problem, which is a well-known NP-complete problem in computer science.

Dynamic Programming

Dynamic programming is a powerful technique for solving optimization problems that exhibit overlapping subproblems and an optimal substructure. It is often used to solve coin change and subset sum problems by breaking them down into simpler subproblems and storing their solutions to avoid redundant computations.

Transcript

00:01 We are asked to use generating functions to find the number of ways to make change for $100.

00:08 In part a, we're told we can use $10, $20, and $50 bills.

00:18 The generating function for the $10 bills, we can make any multiple of $10 with $10 bills.

00:28 And so the generating function is going to be including all powers of x, which are multiples of 10.

00:35 So this is 1 plus x to the 10th, plus x to the 10th.

00:39 20th plus x to the 30th and so on.

00:46 For the $20 bill, we can make any multiple of $20.

00:52 So the generating function is going to include all powers of x, which are divisible by 20.

00:59 So this is 1 plus x to the 20th plus x to the 40th and so on.

01:10 And for the $50 bill, we can make any dollar amount which is divisible by 50, and so it follows is that the generating function will contain any power of x which is divisible by 50.

01:27 So this is 1 plus x to the 50th plus x to the 100th plus x to the 150th and so on.

01:42 And the total generating function is the product of these three generating functions, which is 1 plus x to the 10th plus x to the 20th and so on times 1.

02:01 Plus x to the 20th plus x to the 40th and so on times 1 plus x to the 50th plus x to the 100th and so on.

02:26 And since you're trying to make change for $100, we want to find the coefficient in the expansion of this generating function of x to the 100.

02:36 And so, first of all, we have that we can abbreviate this as 1 plus x to the 10th plus x to the 20th up to and then x to the 100th times 1 plus x to the 20th plus x to the 40th up to x to the 100th.

03:22 Times 1 plus x to the 50th plus x to the 100th and then using a computer algebra system we can expand this sum and we find that the coefficient of x to the 100 is going to be 10 so there are 10 ways to make change in part b we're asked to make change using $5, $10, $20 and $50 bills you already have the generating functions for 10, $20 ,000, and $50 bills, but for $5 bills, we can make any dollar amount, which is a multiple of 5.

04:17 So the generating function will contain any multiple, any power of x, which is divisible by 5.

04:23 So this is 1 plus x to the 5th, plus x to the 10th, and so on.

04:31 And we have a generating function in total is going to be the product of all four generating functions.

04:39 So this is going to be 1 plus x to the 5th plus x to the 10th and so on times 1 plus x to the 10th plus x to the 20th and so on times 1 plus x to the 20th and so on.

05:03 And finally, times 1 plus x to the 50th plus x to the 100th, and so on.

05:17 Because we're looking to make change for $100, you want to find the coefficient of x to the 100.

05:25 To do this, first notice that the sum can be abbreviated if we're only looking for the coefficient of x to 100 as 1 plus x to the 5th plus x to the 10th up to x to the 100th times 1 plus x to the tenth plus x to the 20th up to x to the 100th times one plus x to the 20th plus x to the 40th up to x to 100th and finally times one plus x to the 50th plus x to the 100th and using the computer algebra system you expand this generating function and find the coefficient of x to the 100, a100, is equal to 49 in this case.

06:37 So there's 49 ways to make change.

06:40 In part c, we're asked to make change for $100 using $5 bills, $10 bills, $20 bills, and $50 bills if at least one bill of each denomination is used.

06:58 So the way this is different is we have to have at least one bill of each denomination.

07:06 What this means is you can only have powers of x that are going to be greater than or equal to the bill amount times one.

07:20 And so we have the new generating function for five dollar bills is now going to be x to the fifth plus x to the tenth.

07:32 And so we have the new generating function for five dollar bills.

07:33 And so, on...

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