You and five friends- a total of six parties- plan to meet once per month to have dinner together,with one of you choosing the restaurant each month. Rather than scheduling the entire year in advance, you decide to make it interesting: each month a single six-sided die will be rolled to determine which of you gets to choose the restaurant that month. How likely is it that everyone will have a chance to eat at their own favorite restaurant? That is, what is the probability that over the next 12 months, each of you will have had at least one opportunity to choose where to eat?
Use inclusion-exclusion to derive a formula and compute the exact desired probability .(Note: in practical applications, in many cases, this step is impossible. In other cases,observing empirical behavior like above can provide insight leading to an exact solution.)