Step 1: Generating function:
The count is the coefficient of $x^{12}$ in $(1 + x + \dots + x^7)(1 + x)^{16}$, because the identical balls give a factor $1 + x + \dots + x^7$ and the distinct balls give $(1 + x)^{16}$.
Step 2: Evaluate:
Collecting the coefficient of $x^{12}$ gives ${}^{16}C_{12} + {}^{16}C_{11} + \dots + {}^{16}C_{5} = 62322$.
Checking the options numerically, ${}^{18}C_6 + {}^{18}C_8 = 18564 + 43758 = 62322$. So (A) is correct.
Final Answer:
62322 ways, option (A).
\[ \boxed{{}^{18}C_6 + {}^{18}C_8} \]