How Many Keys?

You are designing a door. For security reasons, you require at least 2 people from a group of people to open the door. For instance, if there are 3 people A, B, and C in a group and you require any 2 and at least 2 to be present to open the door, then you may use 3 locks on the door. Assign person A key 1 and 2, person B key 1 and 3, and person C key 2 and key 3. Consequently, the total number of keys is 2+2+2=6. This is a solution to the (3,2) problem, which means there are 3 people total and you require 2 to open the door. Please note you are free to design the number of locks on the door and assign keys to each person in the group, but assume each lock you use must be unique to the others. Everyone in the group does not need to have the same number of keys. The efficiency is defined as the total number of keys. For instance a solution of 10 keys is better than one of 12 keys.

Find the most efficient solution to the (4,3) problem.

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