Final answer:
The grandfather can be seated in the middle, with his two grandsons seated in 9 different ways to ensure they are not adjacent, and the four granddaughters then have 24 ways to be seated in the remaining places. This results in a total of 216 different ways to arrange the family members for a photograph.
Step-by-step explanation:
Arranging Family Members for a Photograph
When considering the arrangement of a grandfather, two grandsons, and four granddaughters for a photograph with the condition that the grandfather is always in the middle and the two grandsons are never adjacent to each other, we must calculate the possible permutations with restrictions.
First, let's place the grandfather in the middle. There are seven positions in the row (1, 2, 3, G, 5, 6, 7) with the grandfather (G) fixed in the fourth position.
Now, we need to place the two grandsons (GS) in such a way that they are not adjacent to each other, which can be done as follows:
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- One grandson to the left of the grandfather (positions 1, 2, or 3) and one to the right (positions 5, 6, or 7).
This gives us a total of 3 positions on the left and 3 positions on the right, resulting in 3 x 3 = 9 ways to seat the grandsons.
Finally, the four granddaughters (GD) can be seated in the remaining four positions in 4! = 24 different ways. However, this arrangement has to be multiplied by the number of ways the grandsons can be placed:
Total arrangements = 9 ways (GS) x 24 ways (GD) = 216 ways.
Therefore, the grandfather can be seated along with his two grandsons and four granddaughters in 216 different ways under the given conditions.