What is the number of logically possible orders of all six cousins in terms of increasing age?

Correct Option (2)

Based on the information provided, the following age relationships and assignments can be established for the six cousins:

• The oldest cousin, E, is 22 years old.
• The youngest cousin is 17 years old.
• The general age order, from oldest to youngest, is E > A > C > (B or D) > F > (D or B).

Assigning specific ages based on this hierarchy:

• E = 22 years
• A = 21 years
• C = 20 years
• F = 18 years

The remaining cousins, B and D, must occupy the 19-year and 17-year age slots. This leads to two distinct logical arrangements for the increasing order of age:

1. D (17 years) < F (18 years) < B (19 years) < C (20 years) < A (21 years) < E (22 years)
2. B (17 years) < F (18 years) < D (19 years) < C (20 years) < A (21 years) < E (22 years)

Therefore, there are exactly two logically possible orders of the six cousins in terms of increasing age.

Incorrect Options:

Options 1, 3, and 4 are incorrect because the established age relationships and the specific placement of cousins B and D allow for precisely two distinct permutations that satisfy all conditions. There is more than one possible order, eliminating option 1. Similarly, there are not three or four distinct orders, thus eliminating options 3 and 4.