Question

The following 12 adventurers are on a quest to save the world from an evil clown: Terra, Locke, Edgar, Sabin, Shadow, Cyan, Gau, Celes, Relm, Strago, Setzer, and Mog. To avoid detection, the adventurers decide to split up into three four-person parties.

a) How many combinations of adventurers exist for these three parties?

b) Consider that the first party must include Terra, the second party must include Locke, and the third party must include Celes, meaning that these three adventurers cannot share a party with each other. How many party combinations exist with this restriction?

c) With the restriction from part (b) in place, if Terra, Locke, and Celes have their party members selected at random, how likely is it that Terra will be grouped with Edgar, Sabin, and Setzer?

Answer 1

(a)

Number of ways first 4 member team can be chosen	=12*11*10*9/(4*3*2)	495
Number of ways second 4 member team can be chosen	=8*7*6*5/(4*3*2)	70
Number of ways third 4 member team can be chosen	=4*3*2*1/(4*3*2)	1
Total		566

(b)

Number of ways first 4 member team can be chosen	=9*8*7/(3*2)	84
Number of ways second 4 member team can be chosen	=6*5*4/(3*2)	20
Number of ways third 4 member team can be chosen	=3*2*1/(3*2)	1
Total		105

(c) Number of ways condition in (b) holds AND Terra will be grouped with Edgar = 6*5*4/(3*2) + 3*2*1/(3*2) = 21

Hence chance that Terra will be grouped with Edgar, Sabin ans Setzer given condition (b) is true = 21/105 = 0.2