Joey prepares 3 cards for his 3 girlfriends. He addresses 3 corresponding
envelopes. A brown-out suddenly occurred and he hurriedly placed the cards in
the envelope at random. What is the probability that
a. Each card is sent to its proper addressee?
1 way out of 3! or 1/3! or 1/6
b. No card is sent to the proper addressee?
That's 1 out of 3 sub-factorial. But if you haven't studied sub-factorials,
then we have to list the ways they could have all gone wrong. Girlfriend 1
could have been sent card 2 or 3, and in either case there is only one
way the other one could have gone wrong.
gf 1 2 3
card 2 3 1
or
gf 1 2 3
card 3 1 2
[Actually 3 sub-factorial = 3!(1/0!-1/1!+1/2!-1/3!) = 6(1/1-1/1+1/2-1/6) =
6(1/2-1/6) = 6(3/6-1/6) = 6(2/6) = 2]
So there are only 2 ways the cards could have all gone wrong:
2/3! = 2/6 = 1/3
Edwin