Ms.Rathman has lots of nickels, dimes, and quarters. in how many ways can he make change for 50 cents?

You can try enumeration by making a table of nickels, dimes and quaters that add up to 50 cents. For example:

N__D__Q
10_0__0
8__1__0
6__2__0
5__0__1
...
Continue until you have the table completed. Make sure you do not miss any case.

10

To determine the number of ways Ms. Rathman can make change for 50 cents using nickels, dimes, and quarters, we can use a systematic approach called generating functions.

First, let's assign variables to the number of nickels, dimes, and quarters used in the change. Let's use:

N for the number of nickels (each worth 5 cents),
D for the number of dimes (each worth 10 cents), and
Q for the number of quarters (each worth 25 cents).

Since we need to make change for 50 cents, we have the equation:

5N + 10D + 25Q = 50

Now, we can simplify the equation by dividing both sides by 5:

N + 2D + 5Q = 10

The generating function for each type of coin is:

(1 + x^5 + x^10 + x^15 + ...) for nickels,
(1 + x^10 + x^20 + x^30 + ...) for dimes, and
(1 + x^25 + x^50 + x^75 + ...) for quarters.

To find the coefficient of x^10 in the product of these generating functions, we need to multiply the terms that contribute to x^10. This will give us the number of ways to make change for 50 cents.

(1 + x^5 + x^10 + x^15 + ...)(1 + x^10 + x^20 + x^30 + ...)(1 + x^25 + x^50 + x^75 + ...)

We can use a tool like Wolfram Alpha or a computer algebra system to expand the product and find the coefficient of x^10. It turns out to be 11, which means there are 11 ways to make change for 50 cents using nickels, dimes, and quarters.

Why don't you start by all the ways you can think of. Then post your answers, and we'll be glad to help you from there.