The first thing you have to do is come up with a distribution that corresponds to the actual problem/process.
In your problem this is a multivariable hyper-geometric distribution since you are sampling without replacement. However in the case that you have a large sample size, you can approximate this well with a multinomial distribution.
In your probability, you need to find out P(C1 > 0, C2 > 0, C3 > 0, ..., C6 > 0, N > 7).
Hint: With a multinomial, try finding the probability that less than 8 cars pass with all colors appearing once and find 1 - that probability.
Multinomial distribution - Wikipedia, the free encyclopedia
Remember that we are using a approximation if the number of cars is extremely large.