In how many different ways can the set of 10 coins have the total value of 59 cents?

Question

A set of 10 coins  may contain any combonation of pennies nickles dimes quarters and halfdollars.

Answer 1

Let p, n, d, q, h=number of pennies, nickels, dimes, quarters, half-dollar coins respectively.

p+n+d+q+h=10, and p+5n+10d+25q+50h=59.

Start with making up 9¢: 

(1) p=4, n=1 (5 coins used, value 9¢)⇒d+q+h=5 coins, 10d+25q+50h=50¢, so h=0 and d+q=5 and 10d+25q=50, so 2d+5q=10, therefore 2(5-q)+5q=10, 10+3q=10, q=0, so d=5, p=4, n=1, h=q=0.

(2) p=9, h=1 (all 10 coins used)⇒n=d=q=0

Therefore there are only 2 ways to make 59¢ using 10 coins: 5 dimes, 1 nickel, 4 pennies; and 1 half-dollar, 9 pennies.