Freddie has 6 toy cars and 3 toy buses, all different. He chooses 4 toys to take on holiday with him.
(i) Freddie has a total of 9 toys (6 cars + 3 buses). The number of ways to choose 4 toys from 9 is given by the combination formula:
\(\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126\)
(ii) If Freddie's choice must include both his favourite car and his favourite bus, he has already chosen 2 toys. He needs to choose 2 more toys from the remaining 7 toys (5 cars + 2 buses). The number of ways to choose 2 toys from 7 is:
\(\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21\)