jmc

First, count the total number of possible sandwiches. There are four choices for bread, six for meat, and five for cheese, for a total of $4\cdot 6\cdot 5=120$ potential sandwiches. Discarding the ham / cheddar cheese combination will subtract four sandwiches (one for each type of bread). Similarly, discarding the white bread / chicken combination will subtract five sandwiches, one for each kind of cheese. Thus, Al can order $120-4-5=\boxed{111}$ different sandwiches.