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Lift Problem

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    A lift is placed in a building of n+1 floors labelled 0 to n. Before the lift departs the kth floor, the number of people in the lift is \binom{n}{k}. Once people have got off the lift they cannot go back onto it. Initially the lift is at floor 0 and moves upwards. Given that an individual can travel upwards no further than 2 floors (e.g. a person who gets on at floor 1 must leave the lift at the latest at floor 3), find the minimum number of people that use the lift in its first upwards pass.

    I have modelled it for numerical scenarios e.g. 5 floors
    ON OFF
    floor 0 : 1 0
    floor 1 : 3 0
    floor 2 : 3 1
    floor 3 : 1 3
    floor 4 : 0 3

    I notice that it appears to produce another binomial expansion of n-1 but I am not sure how to prove it algebraically.

    Any pointers.
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