So for a group a 7 people, find the probability that all of their birthdays do not occur in the winter. That is, all of their birthdays occur either in the spring, summer or fall. Assume that the probability of being born in each season is equally likely.
So the answer to this one is pretty simple as it is just $\frac{3}{4}^7=0.133$. However, I thought I would try it with a different method. I did it by using the stars and bars counting method.
I counted how many ways there are to arrange 7 people into 3 seasons and also how many ways there are to arrange 7 people into 4 seasons.
Ie. $$P(no\; birthdays\; in\; the\; winter) = \frac{\binom{7+3-1}{3}}{\binom{7+4-1}{4}}=0.4$$
Why is this not getting the same answer?
Best Answer
There are $3^7$ ways to assign $3$ seasons to $7$ people. $\binom{7+3-1}{3}$ counts multisubsets, which assumes that the people can't be distinguished, and isn't appropriate for this problem.