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.

(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

h) number of ways = 5×4×(5)²³ = 4×( 5 )²⁴

Note : tris can be assigned to a faction in five different ways..followed by caleb who can be assigned to a different faction in 4 ways ...while the others 23 candidates can be assigned to their faction in 5 different ways each.

i)putting one candidate each in dauntless and erudite first of all.( two such candidates can be selected in ²⁵C₂ ways )

We are left with 23 candidates who can be alloted to any faction in 5 ways each..

So,

total number of ways to distribute in such a order

= ²⁵C₂ ×( 5 )²³ = 25×24×(5)²³ = 24×(5)²⁵

Note tje superscript show the power raised to 5..