The correct option is B P(E2)=(2m−1)n2mn
Let A={a1,a2,...an}
Let S be the sample space and E1 be the event that Pi∩Pj=ϕ for i≠j
and E2 be the event that P1∩P2∩...∩Pm=ϕ
Therefore numbers of subsets of A=2n
Therefore each P1,P2,...Pm can be selected in 2n ways
∴n(S)= total number of selection of P1,P2,...Pm=(2n)m=2nm
When P1∩P2∩...∩Pm=ϕ
i.e elements of A does not belong to all the subsets.
There are 2m ways on element does not belong to a subset,
on the other hand, there is only one way the elements can belong to the intersection.
Therefore (2m−1) elements does not belong to the intersection.
n(E2)= numbers of favorable ways for all n elements =(2m−1)n
Hence the required probability
P(E2)=n(E2)n(S)=(2m−1)n2nm