Question

Birthday Problem For this problem, assume that a year has $365$days.

(a) How many ways can $18$people have different birthdays?

(b) What is the probability that nobody has the same birthday in a group of $18$people?

(c) What is the probability in a group of $18$people that at least$2$people have the same birthday?

Short answer

The answers are-

(a)

(b)

(c)

Step-by-step solution

Part (a) Step 1. Given information.

Number of people is $18.$

Number of days are$365.$

Part (a) Step 2. Calculation.

In this question $18$birthdays are selected from a possible $365$days, no birthday may repeat itself. The number of ways that this can occur is-

$$\begin{gathered} P(365,18)=\frac{365!}{(365-18)!} \\ =\frac{365!}{347!} \\ =\frac{365\times 364\times 363\times 362........\times 347!}{347!} \\ = \end{gathered}$$