AMC 10 · 2012 · #16
Grade 8 countinglogicPick an answer.
The three pair conditions are all 'at least one' requirements, which are awkward to count head-on. First reframe each song by the set of girls who like it and list the allowed sets. Then, instead of building only the good arrangements, count them by complement: start from every arrangement and use inclusion-exclusion to subtract the ones that miss a required pair.
List each song's possible type
Each item carries one of seven labels.
Naming a song by who likes it turns a fuzzy 'likes and dislikes' story into a clean choice of one label out of seven.
7.SP.C.8Make A Systematic ListRestate the pair rule as coverage
The pair rule says three labels must all appear.
The three separate 'at least one' demands collapse into one picture: all three pair-labels must show up.
7.SP.C.8Change Focus Count The ComplementCount all label assignments first
Counting every assignment is easy.
Counting everything first and then removing the bad cases is easier than building only the good cases directly.
8.EE.A.1Identify SubproblemsSubtract arrangements missing a pair
Subtracting the misses alternates in sign.
Cases missing two pairs get subtracted twice, so inclusion-exclusion adds the overlaps back to keep the count honest.
Cases missing two demands get subtracted twice, so the overlaps have to be added back to keep the count honest.
▸ Why?
A count of overlapping groups double-counts whatever they share, so shared parts are added and removed in turn.
▸ Why?
Counting everything and removing the bad cases is easier than building only the good ones directly.
Compute the final total
The total is 132, choice (B).
Plugging the powers into the inclusion-exclusion formula collapses everything into the single count of good arrangements.
7.NS.A.3Change Focus Count The ComplementWhen a problem demands 'at least one of each,' count every arrangement first and then subtract the ones that leave something out.
- List each song's possible type
- Restate the pair rule as coverage
- Count all label assignments first
- Subtract arrangements missing a pair
- Compute the final total