A machine accepts​ nickels, dimes, and quarters. Exact change is needed. How many ways can a person with four ​nickels, three​ dimes, and two quarters make a 70​-cent purchase from the​ machine?

Question

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

four

​nickels, three​ dimes, and two quarters make a

70​-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.

Answer 1

4N + 3D + 2Q = $1

for 70 cent purchase, you'll left with 30 cents,

either 1 quarter and a 1 nickel 1Q + 1N

3 dimes = 3D

2 dimes and 2 nickels = 2D +2N

1 dime and 4 nickels = 1D + 4N

for each of these 4 ways subtract them from 4N+3D+2Q to get 4 ways to make the purchase

4N+3D+2Q -1Q-1N= 3N+3D+1Q

for 70 cent purchase

use 1 quarter and 3 nickels + 3 dimes

or 2 quarters and 2 dimes

or 2 quarters and 1 dimes + 2 nickels

or 2 quarters and 4 nickels

4 ways, use the last way to have less coins in your pocket left over