If you randomly call $5$ students and each call is independent, what is the probability that no student is called twice? I am thinking about this question and I don't really know how to tackle it.
The problem is:

The class has $30$ students. If you want to call 5 students, each one is independent of the others and randomly with equal probability, what is the probability that none are repeated?

I am thinking as $$\frac{30\cdot 29\cdot 28 \cdot  27\cdot 26}{30^5},$$ but I am not sure it is correct. Thank you so much!
 A: Think about it as a bowl with 30 blue marbles (all the students). Whenever a student is called, we replace a blue marble with a red marble (so this is a student we have already called). Now what is the probability that we don't grab a red marble after 5 times?
So the first attempt, P(blue) is obviously one.
The second attempt, we have P(blue) = $\frac{29}{30}$.
The third, fourth and fifth attempt, we have respectively:
$\frac{28}{30}$, $\frac{27}{30}$, $\frac{26}{30}$
Multiplying these
$1 \times \frac{29}{30} \times \frac{28}{30} \times \frac{27}{30} \times \frac{26}{30}$
we get $\approx$ $70.4$%...
The same will be:
$(\frac{1}{30})^5 \times (30 \times 29 \times 28 \times 27 \times 26) $
So yes, your answer is correct as well.
A: Yes this is correct.
$30 \cdot 29 \cdot 28 \cdot 27 \cdot 26$ or $^{30}P_5$ is the number of ways to permute $5$ students out of $30$ without repetition. $30^5$ is the total number of ways, so the probability is:
$$\frac{30 \cdot 29 \cdot 28 \cdot 27 \cdot 26}{30^5} \approx 0.704.$$
