Five distinct families, each consisting of 3 people are randomly arranged in a straight line. Let X denote the number of families that are sitting next to each other (that is all three people are sitting together). What is E[X] and Var[X]?
There are 15 people.
Number of ways all five families sit together
= 5!*(3!)^5 as there are 5! ways to arrange the families and 3! ways to arrange the members of the family
Number of ways four families sit together
= ways four families sit together and one family does not
= ways five families sit together - ways one family does not
= 5!*(3!)^5 - 5C1(???)
I'm unsure how to calculate the number of ways one family does not sit together. Also is there an easier way to calculate expectation and variance (perhaps using indicator variables) than to find E[X] by finding P(X = n) where n = 1,2,3,4,5? The way I'm trying to solve the problem is pretty tedious