A, B, C, D, E and F are cousins. No two cousins are of the same age, but all have birthdays on the same day of the same month. The youngest is 17 years old and the oldest E is 22 years old. F is somewhere between B and D in age. A is older than B. C is older than D. A is one year older than C.

Which one of the following is possible?

The correct option is F is 18 years old.

Let us break down the given information step-by-step to find the possible age assignments for the cousins: A, B, C, D, E, and F.

1. Determine the Range of Ages:
There are 6 cousins, all with distinct ages. The youngest is 17 years old and the oldest is E, who is 22 years old.
Thus, the set of ages available is: {17, 18, 19, 20, 21, 22}.
We are given that:
$$E=22$$
Therefore, the remaining cousins {A, B, C, D, F} must be assigned ages from the set: {17, 18, 19, 20, 21}.

2. Analyze the Conditions:
- A is one year older than C:
$$A=C+1$$
- C is older than D:
$$C>D$$
- A is older than B:
$$A>B$$
- F is somewhere between B and D in age. This means F's age is between B and D (either $B<F<D$ or $D<F<B$).

3. Test the Option "F is 18 years old":
Let us assume F is 18 years old ($F=18$) and check if we can form a valid sequence of ages.
If $F=18$ and F is between B and D in age, we can assume the ordering:
$$B<F<D$$
Since $F=18$, B must be younger than 18. The only available age younger than 18 is 17.
Thus, we set:
$$B=17$$
With B = 17 and F = 18, D must be older than 18. Let us choose:
$$D=19$$
Now, the remaining available ages for A and C from our set {17, 18, 19, 20, 21} are {20, 21}.
Since A is one year older than C ($A=C+1$), we can assign:
$$C=20$$
$$A=21$$

4. Verification of the Solution:
Let us verify if this age assignment satisfies all the given conditions:
- Ages: B = 17, F = 18, D = 19, C = 20, A = 21, E = 22. (All are distinct and range from 17 to 22. Correct.)
- F (18) is between B (17) and D (19) in age. (Correct, since 17 < 18 < 19.)
- A (21) is older than B (17). (Correct, since 21 > 17.)
- C (20) is older than D (19). (Correct, since 20 > 19.)
- A (21) is one year older than C (20). (Correct, since 21 = 20 + 1.)
Since all conditions are perfectly satisfied, it is indeed possible that F is 18 years old.