Four people ($1,2,3,4$) each pick a different fruit – apple, banana, cherry and date. The person who picked the apple is next to the person who picked the banana and not next to the person who picked the date. The person who picked the cherry is next to the person who picked the date. The correct combination is :

Let's verify the constraints given in the question for the arrangement configuration ($1,2,3,4$):
* Constraint 1: Apple and Banana must be adjacent.
* Constraint 2: Apple and Date cannot be adjacent.
* Constraint 3: Cherry and Date must be adjacent.

Let's evaluate Option C:

* Position $1$: Apple
* Position $2$: Banana
* Position $3$: Date
* Position $4$: Cherry

Verification:

* Position $1$ (Apple) is next to Position $2$ (Banana). (Constraint 1 satisfied)
* Position $1$ (Apple) is next to Banana, which is not next to Position $3$ (Date). (Constraint 2 satisfied)
* Position $4$ (Cherry) is next to Position $3$ (Date). (Constraint 3 satisfied)

Since option C satisfies all conditions, it is the correct combination.