Question 17: Let n and N be positive integers, and let \mathcal{F} be…

Let $n$ and $N$ be positive integers, and let $\mathcal{F}$ be an arbitrary nonempty family of subsets of the universe $\{1, \ldots, n\}$. Suppose each element $x \in \{1, \ldots, n\}$ in the universe receives an integer weight $w(x)$, each of which is chosen independently and uniformly at random from $\{1, \ldots, N\}$. The weight of a set $S$ in $\mathcal{F}$ is defined as

$w(S) = \sum_{x \in S} w(x)$

We want to explore the probability of the following "good" event: there is a unique set in $\mathcal{F}$ that has the minimum weight among all sets of $\mathcal{F}$.

Based on the above data, answer the given subquestions.

Suppose $n = 2, N = 100$ and the randomly assigned weights are:

$w(1) = 25, w(2) = 25$

Consider all non-empty families $\mathcal{F}$ over non-empty subsets of $\{1, 2\}$. There are seven such families. How many of them do not have a unique minimum-weight subset?