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

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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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