Coin Count

Let $x$ , $y$ , and $z$ as the amount of $\$1$ , $\$5$ , and $\$10$ coins. Then make the equations:

$\begin{cases} x+5y+10z=99\\ 3\leq x+y+z\leq 30\\ \end{cases}$

From $x$ that it has 4 possible values: $4$ , $9$ , $14$ , and $19$ . Then make cases to solve:

$\begin{array}{|c|c|c|c|} \hline x & y & z& t\\ \hline 4 & 2t-1 & 10-t & 1\leq t\leq 9\\ \hline 9 & 2t & 9-t & 1\leq t\leq 8\\ \hline 14 & 2t-1 & 9-t & 1\leq t\leq 8\\ \hline 19 & 2t & 8-t & 1\leq t\leq 3\\ \hline \end{array}$

Combine the result, so there are $\boxed{28}$ ways to make $\$99$ .

I used the following code to find the number of ways.

count = 0

For i = 1 To 30

For j = 1 To 20

For k = 1 To 10

If i + j + k <= 30 And i + 5 * j + 10 * k = 99 Then count = count + 1

Next k

Next j

Next i

MsgBox (count)

' Output : 28