Final answer:
The photographer can arrange the 8 people in a row from the family of 12 people in 210 ways when Mom and Dad are in the photo, and in 5040 ways when Mom and Dad are next to each other in the center.
Step-by-step explanation:
In order to determine the number of ways the photographer can arrange 8 people from a family of 12 people, we need to consider the different possible arrangements. Let's look at each scenario:
(a) Mom and Dad are in the photo:
If Mom and Dad are in the photo, then we have to choose 6 more people from the remaining 10 family members to fill the remaining 6 spots in the row. The number of ways to do this is given by the combination formula: C(10, 6) = 210.
(b) Mom and Dad are next to each other in the center:
In this scenario, we can think of Mom and Dad as a single entity. Thus, we have 7 entities to arrange in the row: the couple (Mom and Dad), and the remaining 6 family members. The number of ways to arrange these 7 entities is 7! = 5040.
Therefore, the photographer can arrange the 8 people in a row from the family of 12 people in (a) 210 ways, and (b) 5040 ways.