AMC 10 · 2004 · #12
Grade 7 countingPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A hamburger is really two decisions glued together: how many patties, and which condiments. Those decisions don't interfere with each other, so this is a signal to split the count into subproblems (Tool #7) — count the patty options, count the condiment collections, then multiply. The condiment part itself splits further: each of the 8 condiments is its own separate yes-or-no decision. Listing those choices systematically (Tool #2) shows that each condiment doubles the number of possible collections, which is exactly a power of two. The multiplication counting principle ties it together: when independent choices are made in sequence, the totals multiply.
Split into patties and condiments
Two independent decisions build the burger: patty count and condiment collection. The total is (patty options) times (collections).
When two choices don't affect each other, every option of one can be matched with every option of the other, so the counts multiply.
7.SP.C.8Identify SubproblemsCount the patty options
One, two, or three patties is just a three-item list, so there are exactly 3 ways to pick the meat.
"One, two, or three" is just three items on a list, so the count is three.
7.SP.C.8Make A Systematic ListCount the condiment collections
Each of the 8 condiments is an independent yes-or-no switch, so the collections number 2 to the eighth = 256, none and all included.
Each new condiment offers a fresh in-or-out switch, so every added condiment doubles the number of possible collections.
Each new condiment offers a fresh in-or-out switch, so every added condiment doubles the collections.
▸ Why?
Each switch is flipped without regard to the others, so the counts multiply.
▸ Why?
Repeating the same factor once per condiment is exactly what a power records.
Multiply the two counts
3 patty options times 256 collections gives 768 hamburgers, choice (C) — 256 alone (B) forgets the patties.
Each of the three patty choices can wear any of the 256 condiment collections, so the collections repeat three times over.
5.NBT.B.5Identify SubproblemsWhen separate choices don't affect each other, count each one and multiply — and a yes-or-no switch on each of n items gives 2ⁿ combinations.
- Split into patties and condiments
- Count the patty options
- Count the condiment collections
- Multiply the two counts