Let $D$ denote the number of days in a year, so $D=365$ or $D=366$ depending on how you are counting.

The probability that no one shares the same birthday is the product of the probabilities that the second person doesn't share their birthday with the first $(D-1)/D$ times the probability the third doesn't share with the first two $(D-2)/D$ and so on down the line, until we get $$ \mathbb P(\text{no common birthdays})=\frac{D-1}{D}\cdots \frac{D-n+1}{D}. $$ So the probability that at least two people share a birthday is $1$ minus this.

Let $D$ denote the number of days in a year, so $D=365$ or $D=366$ (or something else) depending on how you are counting (and which planet you are living on).

The probability that no one shares the same birthday is the product of the probabilities that the second person doesn't share their birthday with the first $(D-1)/D$ times the probability the third doesn't share with the first two $(D-2)/D$ and so on down the line, until we get $$ \mathbb P(\text{no common birthdays})=\frac{D-1}{D}\cdots \frac{D-n+1}{D}. $$ So the probability that at least two people share a birthday is $1$ minus this.