Summation and binomial coefficient I'm studying sampling at the moment, but I can't get the passage below:
When $n$ units are sampled from $N$ units without replacement, then each unit of the population
can occur with other units selected out of the remaining $(N - 1)$ units in the population and each unit occurs in $\binom{N-1}{n-1}$ of the $\binom{N}{n}$ possible samples. So
\begin{equation}
    \sum_{i = 1}^{\binom{N}{n}} \left( \sum_{i=1}^n y_i\right) = \binom{N - 1}{n - 1}\sum_{i = 1}^{N} y_i
\end{equation}
Are there any arguments/calculations that can help me understand better the passage above?
 A: The notation in $$\sum_{i = 1}^{\binom{N}{n}} \left( \sum_{i=1}^n y_i\right) = \binom{N - 1}{n - 1}\sum_{i = 1}^{N} y_i$$ is not good, but I think it is possible to find the intended meaning.
You start with a set of $N$ values presented as $y_i$.

*

*The initial $\sum\limits_{i = 1}^{\binom{N}{n}}$ counts over all the different ways of taking a combination of $n$ items from the big set: this is supposed to represent the possibilities in sampling without replacement.


*The $\sum\limits_{i=1}^n y_i$ is then the sum of the $n$ items in each selected combination. I can see no justification for using the same index for the combinations  and each combination's individual items.


*The $\sum\limits_{i=1}^N y_i$ on the right-hand side is simply the sum of all the values in the big set.
The equality is not difficult to justify intuitively: each $y_i$ from the big set can appear in a total of ${\binom{N-1}{n-1}}$ different combinations, since if it is in a combination then the other elements of the combination fill be the $n-1$ space left, taken from the remaining $N-1$ values of the big set.
