Reference no: EM133669331

Question 1: In a magical realm, a grand ball is being organized where different species of magical creatures will attend. There are 12 wizards, 8 elves, and 5 dwarves. A grand dance will feature a sequence of exactly 7 creatures, and the following conditions must be met:

1. The dance sequence must start with a wizard and end with an elf.
2. No more than two creatures of the same species can dance consecutively.
3. At least one dwarf must be included somewhere in the sequence. How many unique dance sequences can be organized for the grand ball?

This problem involves permutations and combinations.

Question 2: A club with 18 members is organizing a fundraising event and needs to form a series of planning committees, each with specific roles and a strict structure. The committee structure is as follows:

1. Each committee must have exactly 5 members.
2. Out of these 5 members, there must be one chairperson and one secretary.
3. The chairperson cannot be the same individual as the secretary.
4. There are 6 members who are senior and experienced, and at least one of them must be either the chairperson or the secretary for each committee.
5. Each committee must include at least 2 members who have not served on a previous committee.

Given that no member can serve in more than one committee, and considering that committees are distinguishable only if their member composition is different (regardless of the roles within the committee) How many unique planning committees can be formed under these conditions?