We consider 7 concerts which can be scheduled throughout a week (7 days). What's the probability that there are at least two concerts on one day if each day is equally probable to be chosen?
Using the inclusion-exclusion principle my approach is as follows:
$$\sum_{k=1}^7{7 \choose k} {13-k \choose7-k}(-1)^{k-1}$$ are all possible ways of scheduling the concerts on 6 days, on 5 days, 4 days and so on...
All possible combinations of scheduling the concerts on all 7 days sum up to: ${13 \choose 7}$. As all combinations are equally probable we have a Laplace Experiment and the probability of having at least two concerts a day is: $$1-\left(\sum_{k=1}^7\frac{{7 \choose k} {13-k \choose7-k}(-1)^{k-1}}{{13 \choose 7}}\right).$$
Is this correct?
I am a little bit confused because the official solution is: $$1-\left(\sum_{k=1}^7{7 \choose k} \left(1-\frac{k}{7}\right)^7(-1)^{k-1}\right)$$ However, this would imply that the concerts are distinguishable. But in my opinion the question does not clearly mention this because we are only interested in the possibility that at least any two concerts should take place on one day. What do you think? Is my reasoning flawed?