Aman has n keys on a key ring, one of which opens the door to his apartment. having celebrated a bit too much one evening, he returns home only to find himself unable to distinguish one key from another. resourceful, he works out a fiendishly clever plan: he will choose a key at random and try it. if it fails to open the door, he will discard it and choose at random one of the remaining n−1 keys, and so on. clearly, the probability that he gains entrance with the first key he selects is 1/n. show that the probability the door opens with the third key he tries is also 1/n.

In year 1, the bennetts' 25-year-old daughter, jane, is a full-time student at an out-of-state university but she plans to return home after the school year ends. in previous years, jane has never worked and her parents have always been able to claim her as a dependent. in year 1, a kind neighbor offers to pay for all of jane's educational and living expenses. which of the following statements is most accurate regarding whether jane's parents would be allowed to claim an exemption for jane in year 1 assuming the neighbor pays for all of jane's support? a.no, jane must include her neighbor's gift as income and thus fails the gross income test for a qualifying relative.b.yes, because she is a full-time student and does not provide more than half of her own support, jane is considered her parent's qualifying child.c.no, jane is too old to be considered a qualifying child and fails the support test of a qualifying relative.d.yes, because she is a student, her absence is considered as "temporary." consequently she meets the residence test and is a considered a qualifying child of the bennetts.