List of all measures

Question

Let Ω be a four-element set (elements do not matter).

How can I find a list of all measures µ: P(Ω)→ N, for which µ(Ω) = 7, where P(Ω) is Power set of Ω.

Answer 1

This is just equivalent to set of all $4$-tuples $(a,b,c,d)$ in $\Bbb N$ such that $a+b+c+d=7$. We only need to assign integer values to the singletons and the measure is determined uniquely. It's quite easy to enumerate these; we will have $\binom{10}{7}$ many of them, by classic combinatiorial facts.