15. A coin collector wants to exchange a 3.50 yuan banknote for 1 fen, 2 fen, and 5 fen coins. He requires the total number of coins to be 150, and each type of coin must be at least 20. Additionally, the number of 5 fen coins must be more than the number of 2 fen coins. Please design an exchange plan accordingly.
15. Let the number of 1 cent, 2 cent, and 5 cent coins be \( x \) coins, \( y \) coins, and \( z \) coins, respectively. According to the problem, we have:
\[
\left\{\begin{array}{l}
x+y+z=150, \\
x+2 y+5 z=350, \\
z>y, \\
x \geqslant 20, y \geqslant 20, z \geqslant 20 .
\end{array}\right.
\]

From (1) and (2), we get:
\[
\left\{\begin{array}{l}
x=3 z-50, \\
y=200-4 z \text{. }
\end{array}\right.
\]

Substituting \( x \) and \( y \) into (3) and (4), we get:
\[
\left\{\begin{array}{l}
3 z-50 \geqslant 20, \\
200-4 z \geqslant 20, \\
z>200-4 z .
\end{array}\right.
\]

Solving these, we get \( 40 < z \leqslant 45 \). Therefore, \( z=41, 42, 43, 44, 45 \).

From this, we can find the corresponding values of \( x \) and \( y \). Thus, we have 5 solutions:
\[
\left\{\begin{array}{l}
x=73, \\
y=36, \\
z=41 ;
\end{array}\left\{\begin{array}{l}
x=76, \\
y=32, \\
z=42 ;
\end{array}\left\{\begin{array}{l}
x=79, \\
y=28, \\
z=43 ;
\end{array}\right.\right.\right.
\]
\[
\left\{\begin{array}{l}
x=82, \\
y=24, \\
z=44 ;
\end{array} \quad \left\{\begin{array}{l}
x=85, \\
y=20 \\
z=45
\end{array}\right.\right.
\]