Answer
See below
Work Step by Step
We have the formula of Permutations with Repetition $\frac{n!}{s_1!.s_2!....s_k!}$
3 German shepherds $s_1=3$
5 collies $s_2=5$
4 cocker spaniels $s_3=4$
3 Corgis $s_4=3$
The total number of dogs is $n=15$
The number of ways to line up the dogs is: $$\frac{15!}{3!.5!.4!.3!}\\=\frac{1307674368000}{6.120.24.6}\\=12612600$$
