Example 1.10 Find the number of 6-permutations of the multiset $M=\{5 \cdot a, 3 \cdot b\}$.
Solution: Let the required number be $N$. Since the 6-subsets of $M$ are as follows: $A_{1}=\{5 \cdot a, 1 \cdot b\}, A_{2}=\{4 \cdot a, 2 \cdot b\}, A_{3}=\{3 \cdot a, 3 \cdot b\}$, and the number of permutations of $A_{1}$ is $\frac{6!}{5!1!}=6$, the number of permutations of $A_{2}$ is $\frac{6!}{4!2!}=15$, and the number of permutations of $A_{3}$ is $\frac{6!}{3!3!}=20$, so by the addition principle, we get
$$
N=6+15+20=41 \text {. }
$$