Undergrad Help with the color of a star cluster

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The discussion revolves around calculating the B-V magnitude of a star cluster using known B-V values for different stellar types. The user has successfully derived a formula for the cluster's magnitude based on stellar types but struggles with applying it to B-V values, resulting in inaccurate outcomes. Attempts to average luminosities and consider the difference in B-V values between star types have not yielded satisfactory results. The user seeks clarification on the correct procedure for calculating the B-V of a cluster from its constituent stars, noting the absence of MB values in the provided data. Assistance is requested to resolve the confusion and demonstrate the necessary calculations.
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TL;DR
Trying to calculate color of cluster from given distribution of stars, can't seem to get it right.
I am working on a problem which asks for the B-V magnitude of a star cluster given N stars of type F and M stars of type K. I have values for B-V for both stellar types. I've derived a general formula for computing the magnitude of the cluster from the magnitudes of each star type, which worked just fine for the first part of the problem (not shown here).

When I try to use the same formula for B-V values, I get an answer that is way off the mark. I then tried averaging the luminosities to get the luminosity of a 'typical' star in the B-V band, and what it's magnitude would be, and got closer to the answer, but not quite there.

Since B-V stands for mB - mV, I thought maybe the question is asking for the difference of the two stellar types' B-V color (subtracting one B-V from the other), which gave me the closest answer, but still .07 magnitude away. Am I missing a basic concept here?

What is the procedure for calculating the B-V of a cluster from the B-V of its constituent stars? Nowhere in the given data are there values for MB, so I'm at a loss. Thanks for any help.
 
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You need to show your work, e.g. the formula derived.
 

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