You must have at least 4 pennies: this will cover items from 1-4 cents.
After that, you need a nickel: you now have items from 1-9 cents covered.
After this: you need a dime: you now have items from 1-19 cents covered.
Let's think about items between 20-24 cents. Adding a quarter would be too much, so we'll add another dime. We now have 1-29 cents covered.
Since we've surpassed 25 cents, we can now add a quarter. We now have everything between 1-54 cents covered.
To cover the rest, we could add a half dollar.
Putting all the above together, we find the minimum required coins to have exact price between 1 cent and 1 dollar is 4 pennies, 1 nickel, 2 dimes, 1 quarter, and 1 half dollar.
To be able to pay for any item worth 1 cent to $1 or 100 cents, the least number of coins required are the following:
1 50 cents coin
1 25 cents coin
2 10 cents coins
1 5 cents coin
4 1 cent coins
This has been calculated considering the fact that for any value below 5 requires the 4 1 cent coins. With the 5 cent coin they can also be used to create all the values between 5 and 10. 2 10 cents coins are required to fill the gap below 25 cents. With the 25 cents, 10 cents, 5 cents and 1 cent coin all values from 1 to 50 can be created. The 50 cents coin allows all values till 100 cents to be created.
The least number of coins required is 9.