Question

In how many ways can a photographer arrange a family consisting of a mother, a father, and 5 children

Answer 1

If you mean in a line, then for the first position you can choose any of the 7. For the second position, you can choose any of the 6 remaining, etc..

\[N=7*6*5*4*3*2*1=7!=5040\]

Answer 2

what if the mother must be next to the father

Answer 3

If we consider the mother and father to be one person there are 6! possible arrangements. However in each of these arrangements the mother can be either to the left or the right of the father. Therefore if the mother must be next to the father the number of possible arrangements is given by
\[2\times6!=you\ can\ calculate\]

Answer 4

I see you already got your answer