Question

In how many different ways a grandfather along with two of his grandsons and four grand daughters 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 1

As there are $6$ descendants they can be arranged in $6!$ ways

Now subtracting in how many ways we can arrange the boys together $=5!\times 2!$ ways$.$

Adding all the ways in the boys are together, but in the middle two possibilities$=4!\times 2!$

$⟹6!-(5!\times 2!)+(4!\times 2!)=528$