Competition · AMC preparation · step 4 of 4
AMC 8 · 2016 · #13
Grade 7 probabilitycountingPick an answer.
AMC 8 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The set is small (6 numbers), so Tool #2 (Systematic List) can enumerate every unordered pair without any formula — easy to count favorable pairs and total pairs by hand. The favorable event collapses to a clean condition: "the pair contains 0," because 0 times anything is 0 and none of the other numbers multiply to 0. Tool #16 (Complement) gives a one-line sanity check: P(product = 0) = 1 - P(0 is NOT chosen), so we can confirm the answer from the opposite direction.
List every pair
List every unordered pair of two different numbers, pairing each number only with those after it so none is missed or repeated.
Listing pairs in a fixed order (smaller number first, then the partner from later in the set) is the Grade 7 "organized list" move for finding sample spaces.
7.SP.C.8Make A Systematic ListCount the total pairs
Count the pairs in the list — the sample space has 15 pairs in all.
Counting the list directly is faster than recalling C(6, 2) at this age, and it matches the enumeration on the page.
7.SP.C.8Make A Systematic ListMark the pairs with product 0
A product is 0 only when a factor is 0, and only 0 qualifies — so the favorable pairs are the 5 pairs containing 0.
The "zero property of multiplication" (0 × n = 0) is a Grade 3 property — applying it filters the list to just the pairs that include 0.
Exactly 5 of the 15 pairs multiply to 0 — precisely the five pairs that include the number 0.
▸ Why?
A pair multiplies to 0 exactly when it contains 0: putting 0 into a pair forces the product to 0, and a pair without 0 can never reach 0.
▸ Why?
Any number multiplied by 0 is 0, so every pair that includes 0 has a product of 0.
▸ Why?
Zero equal groups of a number collect nothing at all, so the total is 0.
▸ Why?
Every other number in the set is nonzero, and multiplying two nonzero numbers keeps the result nonzero, so a pair without 0 never lands on 0.
▸ Why?
Taking a whole, nonzero number of copies of a nonzero amount always builds up some amount, never nothing.
▸ Why?
There are exactly 5 such pairs because 0 can be matched with each of the 5 other numbers, one pair apiece.
▸ Why?
Each of the 5 remaining numbers forms exactly one pair with 0, matching those numbers one-for-one with the favorable pairs.
Form and reduce the probability
Divide favorable by total and reduce: the probability is , choice (D).
Probability = (favorable outcomes) / (total outcomes) when all pairs are equally likely — the Grade 7 uniform-probability model.
7.SP.C.7Make A Systematic ListList the pairs once, find the ones with 0, and divide — Grade 7 probability is enough to crack this AMC 8 problem.
- List every pair
- Count the total pairs
- Mark the pairs with product 0
- Form and reduce the probability
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