If there are exactly $20$ mm's in each packet, then the number of possible combinations equal the number of non-negative integer solutions to the following equation: $$x_1 + x_2 + x_3+x_4+x_5+x_6=20$$

This problem is a basic pattern and the answer is $\binom{20+6-1}{6-1}=\binom{25}{5}$. Similarly, if there are $21$ mm's in a package, the number of possible different combinations equals $\binom{26}{5}$. Hence the answer to the given problem is $$\binom{25}{5}+\binom{26}{5}+\cdots+\binom{45}{5}=\sum_{n=25}^{45}\binom{n}{5}$$

This expression can be further simplified using hockey sticker identity. $$\sum_{n=25}^{45}\binom{n}{5}=\sum_{n=5}^{45}\binom{n}{5}-\sum_{n=5}^{24}\binom{n}{5}=\boxed{\binom{46}{6}-\binom{25}{6}}$$