Dave can select collars for his 5 cats in 15,120 unique ways, by using permutations to calculate the possible selections from 9 different colors without color repetition.
Dave wants to buy a new collar for each of his 5 cats, with no repetitions of colors allowed, from a selection of 9 different colors. To determine the number of possible selections of collars, we can use the concept of permutations since the order in which he chooses the collars matters and we are not allowed repetitions. This is a permutation of 9 objects taken 5 at a time, which is calculated using the formula P(n, k) = n! / (n-k)!, where n represents the total number of objects to choose from and k is the number of objects we want to choose.
Substituting the given numbers, we get P(9, 5) = 9! / (9-5)! = 9! / 4! = (9 × 8 × 7 × 6 × 5 × 4!) / 4! = 9 × 8 × 7 × 6 × 5 = 15,120. Thus, there are 15,120 different selections of collars possible.
Dave has 15,120 unique ways to choose collars for his cats, ensuring each of his 5 cats has a different colored collar.