Ordered triplets of the elements of Lattice

Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram:

For any x, y ∈ L, not necessarily distinct, x ∨ y and x ∧ y are join and meet of x, y respectively. Let L3 = {(x,y,z): x, y, z ∈ L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) ∈ L3 chosen equiprobably satisfies x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z). Then
(A) Pr = 0
(B) Pr = 1
(C) 0 < Pr ≤ 1/5
(D) 1/5 < Pr < 1

Answer

no. of ways of selecting elements in L3 = no. of ways in which we can choose 3 out of 5 elements with repetition allowed
= 5*5*5 = 125

Now, when we take x = t , then given condition satisfies for any y and z , here y and z can be taken in 5*5 = 25 ways.

Take x = r , y = p, z = p, then also condition gets satisfied.

So, 26/125 > 25/125

Therefore, pr > 1/5

Ans is (d)

1Comment
Rameez Raza @rameezraza
24 Aug 2017 05:50 am

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