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In a weighted voting system, players form groups called coalitions. However, a **sequential coalition** considers not only who is in the group but also the order of the players. This order matters and differentiates one coalition from another. For instance, if we have seven players labeled from \(P_1\) to \(P_7\), the way they are arranged in a sequence can form entirely different coalitions.
Simply put:

• The coalition \((P_1, P_2, P_3)\) is different from \((P_2, P_1, P_3)\).
• The "sequence" in which they appear changes everything.

Understanding this concept is crucial, especially in decision-making systems where the sequence reflects power or responsibility. It's about the lineup in which players agree to partake.

The concept of **factorials** plays a key role in calculating the possible sequential coalitions. When dealing with factorials, we look at the product of all positive integers up to a certain number. It's denoted by an exclamation mark (!).
For example:

• \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)

This calculation helps us find all the possible ways seven players can be ordered into a sequential coalition.
Key things to remember about factorials:

• They represent permutations, or the number of ways to arrange items.
• They grow very quickly – 7! is quite a large number at 5040!

In a weighted voting system, factorials allow us to determine how many distinct sequences of arrangements exist.

The **order of players** in a coalition can significantly alter the dynamics and implications of that coalition. This is why in a sequential coalition, changing the order changes the entire scenario.
To illustrate:

• If \(P_7\) is first, the number of ways to arrange the remaining six players is indicated by \(6!\), which equals 720.
• Similarly, if \(P_7\) is the last player, you again have \(6!\) ways to arrange the rest before \(P_7\).

Understanding these arrangements helps in strategy formation and predicting outcomes.
Especially in contexts where order suggests authority or control, recognizing the importance of who goes first or last in a lineup becomes crucial. This is why calculations to exclude a player from a specific position, like not having \(P_1\) first, are necessary, as it helps to map out alternative scenarios. Knowing there's one less usual way allows for diverse strategy in coalition-forming.