Question

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

• A. $\frac{5P_5}{5^5}$
• B. $\frac{7P_5}{7^5}$
• C. $\frac{7C_5}{7^5}$
• D. $\frac{5C_5}{5^5}$

Answer 1

The correct option is B $\frac{7P_5}{7^5}$

Lets say we are assigning birthday's to those $5$ people.
Given no restriction, we can do that in $7^5$ ways.
If we want each one to have a birthday on a different day, then it is as if we are arranging $7$ days among $5$ people.
We can do this in ${}^7{P}_5$ ways.
Hence, the answer.