Farm auction distribution

Relevant wiki: Distinct Objects into Distinct Bins

In this problem, the farming items are modeled as "distinct objects", and the farmers are modeled as "bins."

Let $U$ be the set of all distributions of $7$ distinct objects into $3$ distinct bins. $|U|=3^7=2187$ .

Let $A$ be the set of all distributions in which the 1st bin receives exactly $0$ items, and the $7$ items are distributed among the remaining bins. This is equivalent to the set of all distributions of $7$ distinct objects into $2$ distinct bins. $|A|=2^7=128$

Let $B$ be the set of all distributions in which the 1st bin receives exactly $1$ item, and the remaining $6$ items are distributed among the remaining bins. This is equivalent to the set of all distributions of $6$ objects, chosen from $7$ distinct objects, into $2$ distinct bins. $|B|=\binom{7}{6}2^6=448$ .

$U\setminus(A\cup B)$ is the set of all distributions in which the 1st bin receives at least 2 items.

Note that $A\cap B=\emptyset$ , so $|A\cup B|=|A|+|B|=128+448=576$ .

Then $|U\setminus(A\cup B)|=2187-576=\boxed{1611}$ .