I don't understand your logic - at all.

The typical way this is done is via conditional probability. Basically you look at each person in order and then find the probability of the event not occuring given the information you already know.

So start with person 1: It doesn't matter when their birthday is so they can have any birthday - Probability: 1

Person 2: They just have to not share the same birthday as person 1 - Probability: 364/365

Person 3: They just have to not share the same birthday as person 1 and 2 - Probability: 363/365

...

Person 7: They just have to not share the same birthday as persons 1, 2, 3, 4, 5, 6: Probability - 359/365

Multiply those all together to get the probability that nobody shares the same birthday