Question

Instruction: For this problem, you may leave your answer as as unsimplified expressions with factorials, exponents, binomial coefficients, etc. However, you still need to include a brief justification for your results.

(40 points - parts (a)–(h): 4 points each; part (h): 8 points

In a futuristic dystopian Chicago, society is divided into five factions: Abnegation, Amity, Candor, Dauntless and Erudite. At the age of 16, each person is allowed to choose any faction as their permanent social group at the Choosing Ceremony.

This year, there are 25 candidates at the ceremony, including Tris, Caleb, Christina, and Peter. All 25 candidates get called up one at a time to pledge themselves to the faction that they want to join.

(a) How many possible call orders of all 25 candidates have Tris as the last person called?

(b) How many possible call orders of all 25 candidates have Tris, Caleb, Christina, and Peter (in any order) as the first four candidates called up to pledge their allegiance?

(c) How many ways are there to select 6 candidates to join Dauntless?

(d) How many ways are there to distribute all 25 candidates into factions such that Tris, Christina, and Peter all join Dauntless?

(e) How many ways are there to distribute all 25 candidates into factions such that exactly 5 candidates belong to Erudite?

(f) How many ways are there to distribute all 25 candidates into factions if each of the five factions has room for only five incoming candidates?

(g) For every fixed call order, how many ways are there to distribute all 25 candidates into factions such that nobody joins the same faction as the person called up just before them?

(h) How many ways are there to distribute all 25 candidates into factions such that Tris and Caleb belong to different factions?

(i) How many ways are there to distribute all 25 candidates into factions if there is at least one candidate in Dauntless and at least one candidate in Erudite?

Answer 1