Grandfather-Grandchildren Family Photograph Combinatorics Problem.

Question

In how many different ways a grandfather along with two of his grandsons and four granddaughters can be seated in a line for a photograph so that he is always in the middle and the two grandsons are never adjacent to each other. [Answer] 528

Answer 1

I would say, how many ways can you arrange 6 descendants? $6!$

Subtract from that how many ways can you arrange them with the boys together. $5!\times 2!$

Add back all the ways the boys are together, but in the middle two spots. $4!\times 2!$

$$6!-5!2!+4!2!=528$$

Answer 2

Hint: Defining the pair of positions where the grandsons appear, along with the order of grandsons and granddaughters uniquely specifies an arrangement. There are only $4$ pairs of positions which the grandsons can't occupy.

Answer 3

1. how many ways can you arrange the four girls in a line, just by themselves?
2. How many ways can you arrange the two boys in a line, just by themselves?
3. How many ways can you pick two of the six positions for the grandchildren, regardless of the constraint on placing the boys?
4. How many of those layouts in #3 place the boys adjacent to each other? (remember: if the boys are immediately to the left and right of the grandfather, they're not adjacent)