in a room there are 7 persons. The chance that two of them were born on the same day of the week.

Exactly two were born on the same day:

Let's find the numerator first. Numerator is the total number of favorable events.
We can select 2 persons from 7 persons in $$^7C_2$$ ways and 1 day from the seven days in 7 ways => $$^7C_2$$ * 7
There are 5 persons remaining and 6 days remaining. Each of them was born on a different day.
So, we can select 5 days from 6 days in $$^6C_5$$ ways and arrange the 5 persons in them in 5! ways => $$^6C_5$$*5!
Total number of favorable events = $$^7C_2$$ * 7 * $$^6C_5$$*5!

Total number of events = $$7^7$$

Probability required = $$\frac{^7C_2 * 7 * ^6C_5 * 5!}{7^7}$$