Suppose that in another country all automobile license plates consist of from four to six digits, any digit from 0 to 9 (one possible plate could be 1010 since it has four digits, another plate could be 10101 since it has five digits, and another plate could be 101010 since it has six digits) a. how many license plates are possible if repetition of symbols is allowed? b. how many license plates do not contain any repeated symbol? c. how many license plates have at least one repeated symbol? d. what is the probability that a license plate chosen at random has a repeated symbol?

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.