I am learning Probabilities with the Textbook "Introduction to Probability Models 10th Ed." by Sheldon Ross. At page 74, there is the section covering the distribution of the number of events that occur.
Consider arbitrary events $ A_1,\dots,A_n$. For $1\leq k\leq n$, let $$S_k=\sum_{i_1<\dots< i_k}P(A_{i_1}\cdots A_{i_k})$$ Clearly, this equals the sum of the probabilities of all $\binom{n}{k}$ intersections of $k$ distinct events.
Now consider $$\sum_{i_1<\dots<i_k}\sum_{j\notin \{i1,\dots,i_k\}}P(A_{i_1}\cdots A_{i_k}A_j)$$
The probability of every intersection of $k+1$ distinct events $A_{m_1}\cdots A_{m_{k+1}}$ will appear $\binom{k+1}{k}$ times in this multiple summation. This is so because each choice of $k$ of its indices to play the role of $i_1,\dots,i_k$ and the other to play the role of $j$ results in the addition of the term $P(A_{m_1}\cdots A_{m_{k+1}})$. Hence, \begin{align}\sum_{i_1<\dots< i_k}\sum_{j\notin \{i1,\dots,i_k\}}P(A_{i_1}\cdots A_{i_k}A_j)&=\binom{k+1}{k}\sum_{m_1<\dots< m_{k+1}}P(A_{m_1}\cdots A_{m_{k+1}})\\ &=\binom{k+1}{k}S_{k+1}\end{align}
I cannot understand the last idea. Why is it that in the multiple summation, the probability of every intersection of $k+1$ distinct events $A_{m_1}\cdots A_{m_{k+1}}$ will appear $\binom{k+1}{k}$ times?
For example, if $(m_1, m_2, m_3, m_4) = (1,3,4,6)$, then $P(A_1 A_3 A_4 A_6)$ appears $4$ times in the double summation corresponding to the following choices of $(i_1, i_2, i_3), j$: