There is a rebuttable presumtion that the driver of a vehicle that impacts with the rear of another vehicle is at fault. Under the above scenario both B and C could be held liable for A's injuries and damages depending upon the facts. Seeing that the vehicles were travelling and not standing still B has to allow for a safe distance between it and A. If the accident was such that B was following too closely he/she may be liable even if C caused the chain of events.
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Liability in a crash involving three moving vehicles is not always simple. "B" may be at fault as well. I would hire counsel and likely a lawsuit will be needed to determine percentages of fault.
A needs to sue C and B. The jury will apportion liability between C and B and the damages assessed against them will be split proportionally.
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Without knowing additional information, it appears that vehicle C would have the primary liability. There could be extenuating circumstances that could place some liability on vehicle B as well. More information is needed.
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