Five people born in different months

Text

In a group of $5$ people, assuming the $12$ months are equally likely as a birth month, what is the probability that all $5$ were born in different months?

Solution

Assigning a birth month to each of the $5$ people is like writing a string of $5$ months taken from $12$, with repetition allowed.

• Possible cases: $12^{5} = 248\,832$.
• Favorable cases: all months different, i.e. simple arrangements $12\cdot 11\cdot 10\cdot 9\cdot 8 = 95\,040$.

Hence $P = \frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{12^{5}} = \frac{95\,040}{248\,832} \approx \boxed{0.382}.$