My aunt has seven black cats she alone can distinguish Zeus

My aunt has seven black cats she alone can distinguish: Zeus, Poseidon, Athena, Aphrodite, Ares, Hera, and Apollo . The cats have seven different favorite sorts of food, seven different favorite places to sleep, and seven favorite toys.

1. The cats live in harmony: They exchange their favorite food, places to sleep, and favorite toys among each other every week. However, every sort of food, every place, and every toy always belongs to exactly one cat. How many of such combinations exist in total?

2. Recently, Athena did not feel well and did not like to eat anything. Hence, Aphrodite and Apollo decided to stay with Athena instead of picking an own place to sleep. Aphrodite brought her favorite toy with her and also picked one for Athena, while Apollo was so worried that he was not interested in toys that week. During that time when Athena did not eat, Aphrodite and Apollo did not have an own place to sleep, and Apollo did not have a favorite toy: How many possible combinations existed?

Solution

1 For each of food, place to sleep and toy, there are 7 distinct choices.

Required combinations = 7*7*7 = 7³ = 343.

2 Since Athena did not eat, this is equivalent to 6 sort of food. Similarly, as Apollo was not interested in toys, total of 6 toys and only 5 places as Aphrodite and Apollo decided to stay with athena.

Possible combinations = 6*6*5 = 180.