How to add up cells in a row based on criteria in neighbouring cell

Good afternoon,

I am trying to create a spreadsheet in which I have to add up numbers based on the value in the cell next to them. I have attached a test file so it is easier to understand:

Here the explanation of the sheet:
Cells F, H, J, etc. with the letter A, B, C indicate a cost center
Cells G, I, K, etc. contain the amount of hours worked in that week
Cells C2, D2, E2 contain multipliers for each letter (A, B, C)

What I need now, is a formula in C5 that adds up all the hours with an "A", multiplies it with C2 + all the hours with a "B", multiplies it with D2 + all the hours with a "C", multiplies it with E2.

How can this be done?? Any help is highly appreciated