Problem E

Coin Combinations

In Canada, a \(\$2\) coin is called a toonie, a \(\$1\) coin is called a loonie, and a \(25\)ยข coin is called a quarter. Four quarters have a value of \(\$1\).

How many different combinations of toonies, loonies, and/or quarters have a total value of \(\$100\)?

Note: In solving this problem, it may be helpful to use the fact that the sum of the first \(n\) positive integers is equal to \(\tfrac{n(n+1)}{2}\). That is, \[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}\] For example, the sum of the first \(10\) positive integers is \[1 + 2+3+4+5+6+7+8+9+10 = \frac{10(10+1)}{2} = \frac{10(11)}{2}= 55\]