COINS
by
Teisha, Angie, & Beth
In how many ways can 19 coins equal exactly one dollar?
To begin this problem it is easiest to consider what are not possible solutions. For instance, we know that it is impossible for any one coin type to make sum to a dollar and use exactly 19 coins. Four quarter equal one dollar, but you have only used 4 coins. 10 dimes equal one dollar, but again that is not enough coins. We could use 19 nickels or 19 pennies, but neither equals one dollar.
Next we can try to determine other combinations that cannot be a solution. We know that any solution icluding pennies would have the pennies in multiples of five. It is impossible to get coins to sum up to exactly one dollar if you start with 1, 2, 3, or 4 as your number of pennies.
Then we can begin to make a listing for each type of coin and its possible part in a solution.
List1 (pennies) .......List 2 (nickels) ........List 3(dimes) .........List 4 (quarters) ......List 5 (50 cent pc)
.......5...............................1.............................1..............................1................................1
.......10.............................2.............................2..............................2
.......15.............................3.............................3..............................3
........................................4.............................4
........................................5.............................5
........................................6.............................6
........................................7.............................7
........................................8.............................8
........................................9..............................9
.......................................10
.......................................11
.......................................12
.......................................13
.......................................14
.......................................15
.......................................16
.......................................17
.......................................18
From these lists we can determine that there are 3 x 18 x 9 x 3 x 1 possible combinations of these coins. That, of course, would mean that there are 1458 possible combinations. Although that is true, it doesn't take into account that the combinations have two restrictions on them. The coins must sum to exactly one dollar and there must be exactly 19 coins used.
With that tried, I decided to work using trial and error using the coins I knew were possible. Following are some of the solutions found so far. Let us know if you find other and we will continue to search for solutions as well.
Some solutions:
10 pennies--9 dimes
10 pennies--8 nickels--1 fifty cent piece
10 pennies--6 nickels--1 dime--2 quarters
5 pennies--12 nickels--1 dime--1 quarter
5 pennies--9 nickels--5 dimes
15 pennies--2 nickels--1 quarter--1 fifty cent piece
10 pennies--3 nickels--1 quarter--5 dimes
15 pennies--1 dime--3 quarters
18 nickels--1dime
Return to
Teisha's EMT Problems Page
Return to
Beth's EMT Problems Page
Return to
Angie's page