I need help in writing a formula in cell b7. The formula must look to the right and multiply the nonempty cells by the corresponding value in row 3, and I would like to sum up the results.
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Please see my comment to your original post.
That said, I will try to explain how to approach this as I think you intend. (This solution will be a Google Sheets solution which will not work in Excel.)
The first thing you will need to do is to delete everything from Row 11 down: all of your examples and notes must be deleted for the following proposed formula to work correctly.
Once you have no superfluous data below your main chart, delete everything from B6:B (including the header "Total").
Then, place the following formula in cell B6:
={"TOTAL"; FILTER(MMULT(C7:G*1, TRANSPOSE(C$3:G$3*1)), A7:A<>"")}
This formula will return the header text "TOTAL" (which you can change within the formula itself if you like) followed by the calculation you want for each row where a name is listed in A7:A.
MMULT is a difficult function to explain, but it multiplies one matrix ("grid") or numbers by another matrix ("grid") and returns the sum of all products per row (or per column, depending on how you set it up) —— which is what you are trying to do.
MMULT must have every element of both matrices be a real number. To convert potential nulls to zeroes, you'll see *1 appended to each range (since null times 1 is zero).
This assumes that all data entered into C7:G and C3:G3 will always be either a number or null. If you enter text, you'll throw the formula into an error. If you think accidental text entries in those ranges are possible, use this version instead:
={"TOTAL"; FILTER(MMULT(IFERROR(C7:G*1, ROW(C7:G)*0), TRANSPOSE(IFERROR(C$3:G$3*1, COLUMN(C$3:G$3)*0))), A7:A<>"")}
The extra bits use IFERROR to exchange error-producing entries with zeroes, since MMULT must have every space in both matrices filled with a real number.
