AMC 10 · 2016 · #11

Grade 8 counting

Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region

complementary-countingprinciple-of-inclusion-exclusiondouble-countingsystems-of-equations complementary-countingdouble-counting ↑ Prerequisites: complementary-countingsystems-of-equations

📏 Medium solution 💡 3 insights

Problem

Everyone has one or two of three talents and the missing counts are reported. Count those with exactly two.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Change Focus / Count the Complement

The problem hands over "cannot" numbers, but the question is about talents students do have, so tool #16 (Change Focus / Count the Complement) flips each count first: cannot-sing 42 becomes can-sing 58, and so on. Tool #12 (Draw a Venn Diagram) makes the key structure visible: with the triple-overlap region empty and the outside region empty, every student sits in a one-talent region or a two-talent region. Then tool #4 (Introduce a Variable) names those two group sizes, and tool #15 (Organize Information in More Ways) counts the same camp a second way — by talents instead of by people — which turns the three talent totals into a second equation. Two equations, two unknowns, done. Chasing the six individual regions instead would be far more work, and it is not even possible: the regions are not pinned down by the given data, only their totals are.

1STEP 1

Flip each "cannot" into a "can"

Each report flips into a talent count.

|S| = 100 - 42 = 58, |D| = 100 - 65 = 35, |A| = 100 - 29 = 71

2STEP 2

Everyone has one talent or two

Everyone holds exactly one or two talents.

each student has exactly 1 or exactly 2 talents

3STEP 3

Name the two group sizes

The head count is one equation.

x + y = 100

4STEP 4

Count the same camp by talents

Counting by talent is a second one.

58 + 35 + 71 = 164 ⟹ x + 2y = 164

5STEP 5

Subtract the two equations

Subtracting gives 64, choice (E).

(x + 2y) - (x + y) = 164 - 100 ⟹ y = 64 → (E)

Answer

Build an actual camp and test it. Take 36 students who only act, 29 who sing and dance, 29 who sing and act, and 6 who dance and act; that is 36 + 29 + 29 + 6 = 100 students and nobody has three talents. Singers: 29 + 29 = 58, so 42 cannot sing. Dancers: 29 + 6 = 35, so 65 cannot dance. Actors: 36 + 29 + 6 = 71, so 29 cannot act. All three given numbers come out right, and the number with two talents is 29 + 29 + 6 = 64, matching (E). Note the six individual regions are not forced by the data — other splits also fit — but y = 64 is forced in every one of them, which is why the question only asks for that total. The size is also sensible: 164 talents shared by 100 students means most students are doubly talented, so an answer near 64 fits, while (A) 16 or (B) 25 would be far too few.

💡Key takeaway

Turn every "cannot" into a "can", add the three talent totals to get 164, and since nobody has three talents, the 64 extra counts above 100 are exactly the students with two.

Flip each "cannot" into a "can"
Everyone has one talent or two
Name the two group sizes
Count the same camp by talents
Subtract the two equations

Flip each "cannot" into a "can"

Everyone has one talent or two

Name the two group sizes

Count the same camp by talents

Subtract the two equations

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