Final answer:
The number of different arrangements of the trophies is 5040 with no restrictions and 48 when the football trophies are kept together.
Step-by-step explanation:
(a) To find the number of different arrangements of the trophies with no restrictions, we can use the concept of permutations. The total number of trophies is 7. We can arrange them in 7! (7 factorial) ways, which is equal to 5040.
(b) If the football trophies are to be kept together, we can consider them as one group. We now have 3 groups: the cricket trophies, the group of football trophies, and the swimming trophy. Within each group, the trophies can be arranged in a certain number of ways. The cricket trophies can be arranged in 2! ways, the football trophy group can be arranged in 4! ways, and the swimming trophy can be arranged in 1! way. Multiplying these numbers together, we get the total number of arrangements as 2! * 4! * 1! = 48.