How many different ways can you change a dollar using all 5 kinds of coins in the transaction?

Answer 1

At first look this seems overwhelming, but there are only 2 ways. (see below)

The five coins you are referencing are be half dollar, quarter, dime, nickel & penny; with face-values of 0.50, 0.25, 0.10, 0.05, & 0.01 respectively.

Each coin must be used at least 1 time.

so we have H*0.5 + Q*0.25 + D*0.1 + N*0.05 + P*0.01 = 1

with H Q D N & P are whole numbers greater than zero. Wow! Five variables

But H must equal 1 and only 1, since 2 half dollars would equal 1 dollar and no other coins can be used, similarly Q = 1, because 2 quarters = 50 cents, and you already have 50 cents accounted for. So:

1*0.5 + 1*0.25 + D*0.1 + N*0.05 + P*0.01 = 1, or

D*0.1 + N*0.05 + P*0.01 = 0.25 {now only 3 variables to mess with}

D must equal 1, because if you had 2 dimes (0.20) you could not use the

nickels and the pennies at the same time, so now you have:

0.1 + N*0.05 + P*0.01 = 0.25 ---> N*0.05 + P*0.01 = 0.15

So now the problem is much simpler, with only figuring out how many combinations of nickels and pennies to make up 15 cents.

N can only be either 1 or 2, since 3 nickels = 15 cents and no pennies can be used, so there are only 2 ways that use all 5 and they are:

1 half-dollar, 1 quarter, 1 dime, 2 nickels & 5 pennies.

1 half-dollar, 1 quarter, 1 dime, 1 nickel & 10 pennies.