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Excel – Summing the product of multiple ranges if their sum of different permutations of their row# adds up to a target value

I have a messy Excel formula that I’m trying to figure out. I have data with 2 or more columns, each associated with the probability (labeled p1, p2, p3, etc.) of a particular whole number outcome (marked by the column labeled “row#”). To find the final probability, the probability of the sum of the whole numbers coming up when all the scenarios are combined, I need the sum of the products of all of the probabilities where any combinations of the row# add up to a target value.