Sumif only unique items
 

Hello,
I am trying to use a formula to populate a column that shows how much
inventory(in cost) is attached to each department. It does not matter
if part of that inventory is also for another department. I inserted
the new column next to the criteria:
E F
Hardware
Plumbing
Garden
Home Office



Here is my range of data.
A B C D
Hardware Washer $3.33 5
Plumbing Screw $.56 4
Hardware Screw $.56 4
Hardware Washer $3.33 5

Column A is the department who ordered the part, B is what they
ordered, C is the cost, and D is how many we have in inventory.

What I want is column F to multiply column C and D and put the total
there if Column A = Column E, and if it is the first instance of the
item in column B.

This is how it should look:
A B
Hardware ($3.33*5)+($.56*4)
Plumbing $.56*4
Garden
Home Office

Any help would be appreciated.