Final answer:
The number of arrangements that begin and end with a parent when organizing a family photo of a mother, father, and six children is 1440, which is calculated by 2 (permutations of the two parents) times 6 factorial (arrangements of the six children).
Step-by-step explanation:
To determine the number of arrangements of a family for a group photo where the arrangements must begin and end with a parent, we need to consider permutation principles. The family consists of a mother, father, and six children, making a total of eight individuals. However, since we seek the number of arrangements with a parent at both the beginning and the end, we only need to arrange the six children.
The number of ways to arrange the six children is the factorial of 6, which is 6! (6 factorial). The calculation is 6 x 5 x 4 x 3 x 2 x 1, which equals 720.
However, since we can have either parent at the beginning and the other at the end, we need to multiply the arrangement of the children by the permutation of the two parents. There are 2 choices for the first position (mother or father) and 1 choice for the last position (whichever parent is not in the first position).
The total number of arrangements is therefore 2 parents permutations × 6 children factorial, which equals 2 × 720 = 1440. But this is not one of the options provided, so we should double-check our calculation or assumptions.
Upon double-checking, we realize we made a mistake in the initial calculation. There are indeed two options for the first position (either parent) and also two options for the last position (whichever parent is not in the first position), but once we have chosen the first parent, there is only 1 choice left for the last position, so we simply need to multiply the number of arrangements of the six children by 2, because the two parents can be switched in the beginning and the end.
This makes the calculation 2 × 6! = 2 × (720) = 1440. Now, this is the correct number of arrangements.