Aspire Faculty ID #18502 · Topic: UGC NET Computer Science Dec 2022 Shift I (Paper II) · Just now
UGC NET Computer Science Dec 2022 Shift I (Paper II)

Seven people enter a lift. The lift stops only at three floors which are unspecified. At each of the three floors, no one enters the lift, but at least one person leaves the lift. After the three floor stops, the lift is empty. In how many ways this can happen?

Solution

Each of the $7$ people can leave at any one of the $3$ stops. Total ways without restriction: $3^7=2187$ But each stop must have at least one person leaving. Using inclusion-exclusion: Required ways $=3^7-\binom{3}{1}2^7+\binom{3}{2}1^7$ $=2187-3(128)+3(1)$ $=2187-384+3$ $=1806$
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