Question

Assuming equal probability for the days, the probability that the birthdays of seven people will fall on seven different days of the week is

• A. $\frac{1}{7^6}$
• B. $\frac{{}^7{C}_7}{7^7}$
• C. $\frac{{}^7{P}_7}{7^7}$
• D. $\frac{1}{6^7}$

Answer 1

The correct option is C $\frac{{}^7{P}_7}{7^7}$

Lets say we are assigning birthday's to those 7 people.
Given no restriction, we can do that in $7^7$ ways.
If we want each one to have a birthday on a different day, then it is as if we are distributing $7$ days to $7$ people.
We can do this in ${}^7{P}_7$ ways.
Hence, the answer.