Four mathematicians, three physicists, and an engineer are to be seated equally spaced around a circular table. How many different arrangements are possible if the mathematicians must all sit together (in four consecutive seats) and the physicists must all sit together (in three consecutive seats)? (As usual, two arrangements are identical if one is a rotation of the other.)
How many ways can 4 mathematicians sit in a row? 4!
How many ways can 3 physicists sit in a row? 3!
Seat the engineer.
the engineer can have maths people on his left or right, that is 2 ways.
Multiply these three numbers together and you will have your answer.