+41 Let \(S ={ 1,2,3,...,n}\). Three subsets \(A,B,C\) of \(S\) are chosen at random.
(a) Find the probability that \(A \cup B \cup C = S.\)
(b) Find the probability that \(A⊆B⊆C\).
Thank you!
(a) There are 3^n ways of distributing the elements among A, B, and C, and there are 2^n ways of choosing each subset, so the probability is 3^n/(2^n*2^n*2^n) = 3^n/8^n.
(b) There are 2^n ways of choosing C, then 2^(n - 1) ways of choosing B, then 2^(n - 2) ways of choosing A, so the probability is 2^n*2^(n - 1)*2^(n - 2)/(2^n*2^n*2^n) = 1/8.
