The question is the first question in Harvard STAT110:
For a group of 7 people, find the probability that all 4 seasons
(winter, spring, summer, fall) occur at least once each among their
birthdays, assuming that all seasons are equally likely.
The instructor used inclusion-exclusion method to solve the problem, I didn't manage to solve by this method and tried another approach to solve the problem, but the final answer was wrong, here is my answer:
I give every season letter S,W,F,P(spring) and assumed that the pattern (SWFP),which contains all seasons, at the beginning of the 7 letters pattern, so the remain letters are 3 letters and can be repeated so we choose 3 out of 4 and allow repeating $$\binom{n+r-1}{r} \rightarrow \binom{3+4-1}{3} = \binom{6}{3} = 20 $$
and that was the numerator and the denominator is choosing 7 out of 4 with repetition : $$\binom{n+r-1}{r} \rightarrow \binom{7+4-1}{7} = \binom{10}{7} = 120 $$
so the final answer is $\frac{20}{120}=\frac{1}{6}$ which is wrong, can anyone tell me why my answer is wrong ? ( The correct answer is $\approx 0.513$ )
Best Answer
In short: The order matters because the people are distinguishable.
The formula $$ \binom{n+r-1}{r} $$ counts the number of multi-subsets of size $r$ from a set of size $n$. The formula $$ n^r $$ counts the number of strings of length $r$ from an alphabet of size $n$. You need the latter: If you want to know what possible arrangements there are of the months in which 7 people have birthdays, then each arrangement looks like $$ "WSFPFSS" $$ a string of length 7 from an alphabet of 4 letters. Each such arrangement is not uniquely specified by a multi-subset like: $$ "\{ W,S,F,P,F,S,S\}, $$ because you cannot see which person is born in which month.