Lift Problem

Let's call the floors A, B and C and number the persons from 1 to 7. Then, we have to count the number of maps $f:\{1,2,3,4,5,6,7\}\to\{A,B,C\}$ such that each value A, B and C is taken at least once (i.e., surjective functions). Using inclusion-exclusion principle, the total number of maps is $3^7$ , those which avoid one specific value are $2^7$ , those which avoid two specific values are only one (the constant function taking the third value). Since there are 3 subsets of $\{A,B,C\}$ with 1 element and 3 subsets with 2 elements, we get that the number is $3^7-3\cdot 2^7+3\cdot 1=\boxed{1806}.$

Another method: 7 can be split into 3 non-zero parts in the following ways:

• 1, 1, 5 -->7 * 6 * 3!/2 = 126
• 1, 2, 4 --> 7 * 6c2 * 3! = 630
• 1, 3, 3 --> 7 * 6c3 * 3!/2 = 420
• 2, 2, 3 --> 7c2 * 5c2 * 3!/2 = 630

126 + 630 + 420 + 630 = 1806