First, what coins are available:
Penny
Nickel
Dime
Quarter
Half Dollar
Evaluate the coins, starting with the largest denomination:
How many half dollars can we have and not equal $1? 1 half dollar
If we have 1 half dollar, how many quarters can we have? 1 quarter
Because any more quarters, you could give 2 quarters and 1 half dollar
Now we have 1 half dollar and 1 quarter? How many dimes can we have? 4 dimes
Now we have 1 half dollar, 1 quarter, and 4 dimes.
How many nickels can we have? 0 nickels
With 1 nickel: 1 half dollar, 1 quarter, 2 dimes, and 1 nickel = $1
With 2 nickels: 1 half dollar, 4 dimes, and 2 nickels = $1
That means we can only have 4 pennies, because 5 pennies equals 1 nickel
1 half dollar + 1 quarter + 4 dimes + 0 nickels + 4 pennies = $1.19