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 $\approx 0.513$ )

  • 2
    $\begingroup$ You assume all combinations are equally likely when considered unordered, but this is not the case. For example if there were 2 people and 2 seasons then there are (2+2-1)C2 = 3 different possibilities: both season 1, both season 2, or one of each. But the one of each outcome is twice as likely as the other two. $\endgroup$ – Jacob FG Jul 7 '20 at 9:34
  • $\begingroup$ Do you mean that the order does matter ? $\endgroup$ – mathematishan Jul 7 '20 at 9:46
  • 1
    $\begingroup$ Yes, if you want every possiblity to be equally likely, you must take order into account. $\endgroup$ – Jacob FG Jul 8 '20 at 12:16

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.

  • $\begingroup$ if this is the case the problem is unsolvable by using this way $\endgroup$ – mathematishan Jul 7 '20 at 13:39

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