A is a set containing n elements. A subset P of A is chosen at random. The set A is reconstructed by replacing the elements of the subset of P. A subset Q of A is again chosen at random. Find the probability that P and Q have no common elements.
I tried to calculate in this way :
In set P we can have no element i.e.Φ, 1 element, 2 elements, …… upto n elements. If we have no element in P, we will leave by all the elements and number of set Q formed by those elements will have no common element in common with P. Similarly, it there are r elements in P we are left with rest of (n – r) element to form Q, satisfying the condition that P and Q should be disjoint.
Now, my confusion is how can I find the Total number of ways in which we can form P and Q?
Best Answer
Each element of $A$ has 3 choices- it can either be a part of $P$, of $Q$, or neither of the two, since it cannot be a part of both.
Hence by multiplication rule, for $n$ elements, total number of ways to divide the elements = $3^n$.