From a pool of 10 archers forming a team, if the total number of combinations is 120, which factor can represent the number of ways to assign leadership titles?

To find the number of ways to assign leadership titles among a group of 10 archers, it's important to consider the total combinations of selecting leaders and how those selections relate to permutations.

Given that the total number of combinations of choosing a subset of archers is 120, this can be linked to the mathematical representation of combinations where the formula is given by:

[

C(n, r) = \frac{n!}{r!(n - r)!}

]

In the context of this problem, "n" represents the total number of archers (10), and "C(n, r)" refers to how many ways a certain number of archers can be chosen for leadership roles. If we assume that two specific archers are assigned as leaders, the equation can help us compute the number of ways to select these leaders from the total.

However, to assign leadership titles effectively, we may have to consider not just selecting leaders, but also the order in which they are assigned since different titles may apply to the same group of selected leaders. This shifts our focus to permutations, which take into account the specific arrangement of entities.

In this context, if we denote that there are 2 leadership roles (for example) among 10 arch