This is Self Test Exercise 7.8 of Sheldon's A First Course in Probability. I'm lost at what the question is asking, and illustrate this with an example.
Suppose $r = 10$. That means that there are $10$ families in the arriving plane (OK). The confusion starts from the next sentence. "A total of $n_j$ of these families have checked in a total of $j$ pieces ... OK, so $n_{10} = 10$, or each family brings one piece of luggage? But then you're also telling that
$$\sum_{j} n_j = r$$
which leaves me just as confused, because for one I don't know the upper limit (the most probable guess would be r), and I don't think the summation makes sense in that case. (Bing Chat BTW says that "The sum of all the checked-in luggage is equal to the total number of families: ($\sum nj = r$)." which actually makes sense, but I'm not sure the question implies as such, and contradicts the next sentence as they say that $N$ pieces of luggage comes out?)
As a result, I need some help with trying to understand what the question is asking, and would appreciate assistance. Thanks in advance.
As I understand it, $n_j$ represents the number of families who have checked in $j$ baggages each. For example, if there are $5$ families who have each checked in a total of $2$ baggages, $j = 2$ and $n_2 = 5$.
It's clear that when you take the sum of $n_i$, it'll give you the total number of families. So
$$\Sigma{n_j} = r$$
The total number of checked in baggages is given by $\Sigma(j \cdot n_j)$ and this is represented by $N$. Also, this is not necessarily equal to $n$.
For example, there are $5$ families with $2, 3, 4, 4, 4$ checked in baggages respectively. So, $n_1 = 0$, $n_2 = 1$, $n_3 = 1$, $n_4 = 3$. $\Sigma{n_j} = 5$, total number of families. And $\Sigma(j \cdot n_j) = 17$, the total number of checked in baggages.
I would say, perhaps the question should have included each and said -
... A total of $n_j$ of these families have checked in a total of $j$ pieces of luggage each...