A class has 25 students. Assume that each student's birthday is equally likely to be any of 365…

Step 1. All different. The probability is $\dfrac{365 \times 364 \times \cdots \times 341}{365^{25}} = \dfrac{^{365}P_{25}}{365^{25}} \approx 0.4313$.

Step 2. A shared birthday. The complement is $1 - 0.4313 = 0.5687$.

Step 3. The smallest class. The chance of a shared birthday is about $0.891$ for $40$ students and about $0.903$ for $41$ students. So $41$ is the smallest class for which it is more than $0.9$.