I'm having trouble on deciding when it's appropriate to use the stars and bars method for enumeration. For example, the following question:
Suppose each student in a 50-student class selects a number 1 through 10 with uniform probability, that is each number 1 − 10 is equally likely to be chosen by each student. What is the probability that exactly 6 students choose 1?
Would stars and bars be a valid approach here?
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$\begingroup$Does the following ring a bell?
A certain experiment with a success chance of ${1\over10}$ is performed $50$ times independently. What is the probability that we see exactly $6$ successes?
$\endgroup$ $\begingroup$Stars and bars is not the correct approach as you are distributing distinct objects (students) to distinct items (numbers). The sample space consists of tuples $(x_1, x_2,\dotsc,x_{50})$ where $1\leq x_i\leq 10$ with equal probability attached to each of the $10^{50}$ sample points. The number of such tuples with exactly $6$ ones is $\binom{50}{6}9^{44}$ (choose which $6$ of the positions to have ones and the rest of the positions can have any other number other than $1$). Hence the probability is $$ \frac{\binom{50}{6}9^{44}}{10^{50}}=\binom{50}{6}\left(\frac{1}{10}\right)^6 \left(\frac{9}{10}\right)^{44}. $$
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