A collection of coins includes only pennies (1申), nickels (5థ), dimes (10థ) and quarters (25)). Using the coins in this collection, it is possible to create any amount of money less than one dollar (100థ). What is the smallest possible number of coins in the collection?
(A) 10
(B) 7
(C) 11
(D) 13
(E) 12
We first consider the minimum number of coins needed to create 99 , the greatest amount that we are required to create.

We can create $75 \%$ with a minimum number of coins by placing three quarters in the collection. Once at 75 \& , we need $99-75=24$ \& more to reach 99 \&.

To create $24 \phi$ with a minimum number of coins we use two dimes and 4 pennies.

Therefore, the minimum number of coins required to create 99 क is 3 quarters, 2 dimes and 4 pennies, or 9 total coins in the collection.

In fact, this is the only group of 9 coins that gives exactly 99 .

To see this, consider that we must have the 4 pennies and the 3 quarters, but these 7 coins only give 79 \&.

We now need exactly 20 a and have only two coins left to include in our collection.

Therefore the two remaining coins must both be dimes, and thus the only group of 9 coins that gives exactly 99p is 3 quarters, 2 dimes and 4 pennies.

We now attempt to check if all other amounts of money less than one dollar can be created using only these 9 coins.

Using the 4 pennies, we can create each of the amounts $1 \phi, 2 \phi, 3 \phi$, and $4 \phi$.

However, we don't have a way to create 5 \& using 3 quarters, 2 dimes and 4 pennies.

Thus, at least one more coin is needed.

It is clear that if the 10th coin that we add to the collection is a penny, then we will have 5 pennies and thus be able to create $5 \mathrm{q}$.

However, with 3 quarters, 2 dimes and 5 pennies, we won't be able to create 9 .

If instead of a penny we add a nickel as our 10th coin in the collection, then we will have 3 quarters, 2 dimes, 1 nickel and 4 pennies.

Obviously then we can create 5 using the 1 nickel.

In fact, we are now able to create the following sums of money:

- any amount from 1 \& to 4 \& using 4 pennies;
- any amount from $5 \mathrm{p}$ to $9 \mathrm{c}$ using 1 nickel and 4 pennies;
- any amount from 10 to 14 \& using 1 dime and 4 pennies;
- any amount from 15 to 19 using 1 dime, 1 nickel and 4 pennies;
- any amount from 20 to 24 c using 2 dimes and 4 pennies.

If in addition to the amounts from $1 \phi$ to 24 we use a quarter, we will be able to create any amount from 25 to 49 .

Similarly, using 2 quarters we can create any amount from 50 \& to 74 \&.

Finally, using 3 quarters we can create any amount from 75 t to 99 ф.

Therefore, the smallest number of coins needed to create any amount of money less than one dollar is 10 .

Can you verify that there is a different combination of 10 coins with which we can create any amount of money less than one dollar?

ANSWER: (A)