The puppies have different colored noses because of reccesive and dominant genotypes.
Genotypes are represented by letters, capital being dominant and lowercase being recessive.
For example, lets say t = pink nose and T = black nose.
If the parents have black noses, then they will pass on T genotypes.
But, they also have hidden pink nose genotypes.
These hidden genotypes are passed on to the puppies, one of the puppies gets more black nose genotypes.
If a puppy gets tt, they will have a pink nose.
If they get TT, they will have a black nose
If they get Tt, they will also have a black nose.
Because the black nose is dominant, its more likley for the puppies to end up with a black nose.