In the sumif function, can the range value be set to multiple columns?

I have one spreadsheet where I have parts listed for various assemblies. Each row in a different assembly and there are a variety of parts listed for each along that row. Each separate assembly (row) may use some of the same parts from other assemblies, or may not. In the fist column of the row is the total quantity of that assembly

In another spreadsheet, I want to make a bill of material of all the parts needed for all assemblies. I am doing this by using the sumif function to add the quantity if the referenced part appears in the parts list for that assembly. However, the individual part may appear in any of the several columns.

I want to make the range variable in the sumif function reference multiple columns but when I try to set it to multiple columns I get an error.

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