Answer
$P(10~people~have~10~different~birthdays)\approx0.8831$
Work Step by Step
A year has 365 days. So, $N(S)=365$
First person : 365 days.
Second person : 364 days.
Third person : 363 days.
Fourth person : 362 days.
Fifth person : 361 days.
Sixth person : 360 days.
Seventh person : 359 days.
Eighth person : 358 days.
Nineth person : 357 days.
Tenth person : 356 days.
$P(10~people~have~10~different~birthdays)=P(First~person)\times P(Second~person)\times P(Third~person)\times P(Fourth~person)\times P(Fifth~person)\times P(Sixth~person)\times P(Seventh~person)\times P(Eighth~person)\times P(Nineth~person)\times P(Tenth~person)=\frac{365}{365}\times\frac{364}{365}\times\frac{363}{365}\times\frac{362}{365}\times\frac{361}{365}\times\frac{360}{365}\times\frac{359}{365}\times\frac{358}{365}\times\frac{357}{365}\times\frac{356}{365}\approx0.8831$
