How Many Ways to Make 40 Cents with 4 Nickels, 3 Dimes, and 2 Quarters?

Math Problem Statement

A vending machine accepts nickels, dimes, and quarters. Exact change is needed to make a purchase. How many ways can a person with

fourfour

nickels, three dimes, and two quarters make a

4040-cent

purchase from the machine? For this problem, treat all coins within a particular value as if they are the same. That is, do not count the number of ways that a single coin (a nickel, a dime, or a quarter) could be chosen.

Solution

We are tasked with determining how many ways a person can make a 40-cent purchase using 4 nickels, 3 dimes, and 2 quarters. Let's break this down step by step.

1. Define the Variables

Let $n$ represent the number of nickels used.
Let $d$ represent the number of dimes used.
Let $q$ represent the number of quarters

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Coin Combinations
Diophantine Equations

Formulas

n + 5d + 10q = 40 (where n = number of nickels, d = number of dimes, q = number of quarters)

Theorems

Combinatorial counting methods
Non-negative integer solutions to linear equations

Suitable Grade Level

Grades 8-10

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