Final answer:
The question involves calculating the number of ways to make pairs of dancers under specific conditions using combinations in mathematics. Each situation is evaluated based on the roles of leads and follows and special pairing conditions provided.
Step-by-step explanation:
The task given is to pair up 15 dancers in different scenarios with specific conditions. Here are the solutions:
- With 4 leads and 11 follows, the number of ways to pair them is calculated by choosing 4 follows from 11 to be paired with the 4 leads. This can be done in combination ways, denoted as 11C4 or choosing 4 from 11. The formula for combinations is C(n, k) = n! / (k! (n - k)!), where n is the total number of items to choose from, and k is the number of items to choose.
- With 8 leads and 7 follows, where Percy only dances with Annabeth, we first pair up Percy and Annabeth. Then, we have 7 leads and 6 follows left to pair up, which can also be done in combination ways, denoted as 7C6.
- With 7 leads, 7 follows, and Preeti who can be either, we have 8 potential leads and 8 potential follows. After choosing a role for Preeti, we then need to pair up the remaining 7 pairs, which can again be handled with the concept of combinations.
The calculations of combinations will provide the number of ways to make the pairs according to the given conditions.