Probability of exact event occurring? Going through a discrete math textbook on my own time with no answer key and just want to check my work here and see if my thinking is correct. Lets say we have 20 boxes and 3 marbles. We choose 1 box at a time, uniformly at random, to drop a single marble into. What is the probability that exactly 2 boxes will have marbles? In other words, what is the probability that, out of 3 marbles, exactly 2 land in the same bucket?
So here is my thinking:
1.) prob of choosing a specific box $\frac{1}{20}$
2.) prob of second marble going in that same box $(\frac{1}{20})^2$
3.) prob of third marble going to any other open box $\frac{19}{20}$
4.) result: $\left(\frac{1}{20}\right)^2 \cdot\frac{19}{20}$ right?

Probability all three same box $=(\frac{1}{20})^2=\frac{1}{400}$
Probability all three in different boxes $=\frac{19\times 18}{20^2}=\frac{342}{400}$
Probability of exactly two in one box $=(1-$ above summed) $=\frac{57}{400}$.
