Answer to Question #211335 in Discrete Mathematics for Naresh

Question

a)          Freddie has 6 toys cars and 3 toy buses, all different. 

i)              Freddie arranges these 9 toys in a line. Find the number of possible arrangements 

if the buses are all next to each other.

Answer 1

Since the buses are all next to each other, we can cover them by a box and consider this box as a kind of toy car. Then, as we consider, Freddie has 7 toy cars, and therefore, there exists 7!=5040 different ways to arrange them in a line. After doing that, it remains for Freddie to arrange the 3 toy buses under the box. This gives him 3!=6 ways per each arrangement of the toy cars.

The total number of possible arrangements is $7!\cdot 3!=5040\cdot 6=30240$.