There are various ways to make $\$ 207$ using only $\$ 2$ coins and $\$ 5$ bills. One such way is using one $\$ 2$ coin and forty-one $\$ 5$ bills. Including this way, in how many different ways can $\$ 207$ be made using only $\$ 2$ coins and $\$ 5$ bills?
(A) 9
(B) 10
(C) 19
(D) 41
(E) 21
Solution 1

Using $\$ 5$ bills, any amount of money that is a multiple of 5 (that is, ending in a 5 or a 0 ) can be made.

In order to get to $\$ 207$ from a multiple of 5 using only $\$ 2$ coins, the multiple of $\$ 5$ must end in a 5. (If it ended in a 0 , adding $\$ 2$ coins would still give an amount of money that was an even integer, and so couldn't be $\$ 207$.)

Also, from any amount of money ending in a 5 that is less than $\$ 207$, enough $\$ 2$ coins can always be added to get to $\$ 207$.

The positive multiples of 5 ending in a 5 that are less than 207 are $5,15,25, \ldots, 195,205$. An easy way to count the numbers in this list is to remove the units digits (that is, the $5 \mathrm{~s}$ ) leaving $0,1,2, \ldots, 19,20$; there are 21 numbers in this list.

These are the only 21 multiples of 5 from which we can use $\$ 2$ coins to get to $\$ 207$. So there are 21 different ways to make $\$ 207$.

## Solution 2

We are told that $1 \$ 2$ coin and 41 bills make $\$ 207$. We cannot use fewer $\$ 2$ coins, since $0 \$ 2$ coins would not work, so we can only use more $\$ 2$ coins.

To do this, we need to "make change" - that is, trade $\$ 5$ bills for $\$ 2$ coins. We cannot trade 1 $\$ 5$ bill for $\$ 2$ coins, since 5 is not even. But we can trade $2 \$ 5$ bills for $5 \$ 2$ coins, since each is worth $\$ 10$.

Making this trade once gets $6 \$ 2$ coins and $39 \$ 5$ bills.

Making this trade again get $11 \$ 2$ coins and $37 \$ 5$ bills.

We can continue to do this trade until we have only $1 \$ 5$ bill remaining (and so $\$ 202$ in $\$ 2$ coins, or 101 coins).

So the possible numbers of $\$ 5$ bills are $41,39,37, \ldots, 3,1$. These are all of the odd numbers from 1 to 41 . We can quickly count these to get 21 possible numbers of $\$ 5$ bills and so 21 possible ways to make $\$ 207$.

ANSWER: (E)