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 test the conditions against the given options to find the correct linear fruit order ($1,2,3,4$):
Conditions:

1. Apple is next to Banana ($\text{Apple},\text{Banana}$ or $\text{Banana},\text{Apple}$ are adjacent).
2. Apple is NOT next to Date.
3. Cherry is next to Date ($\text{Cherry},\text{Date}$ or $\text{Date},\text{Cherry}$ are adjacent).

Evaluating Option C:

* Position $1$: Apple
* Position $2$: Banana
* Position $3$: Date
* Position $4$: Cherry
* Condition 1 check: Apple ($1$) is next to Banana ($2$). (True)
* Condition 2 check: Apple ($1$) is next to Banana, not Date ($3$). (True)
* Condition 3 check: Cherry ($4$) is next to Date ($3$). (True)

All conditions are perfectly satisfied by option C.