AMC 10 · 2017 · #13

Grade 6 arithmetic
Archetype Overlap / Complement Counting └ Sub-archetype Two-Set Inclusion-Exclusion: Solve for Missing Region
principle-of-inclusion-exclusionset-partitiondouble-counting double-countingconvert-to-algebra ↑ Prerequisites: principle-of-inclusion-exclusion
📏 Medium solution 💡 2 insights
Problem
There are 20 students, and every student takes at least one of three classes: yoga, bridge, or painting. The class sizes are 10 (yoga), 13 (bridge), and 9 (painting). Exactly 9 students take at least two of the classes. Find how many students take all three.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Tool #16 (Change Focus): instead of tracking students, I count class-seats — the total of all three class sizes — because that total quietly counts a multi-class student more than once, and those repeats are exactly what the question is about. Tool #4 (Introduce a Variable): I let c stand for the unknown number of triple-takers so I can express the repeat seats in terms of c. Tool #13 (Convert to Algebra): the seat count becomes one short equation that I solve for c.

1STEP 1

Count total class seats

Add the three class sizes: 10+13+9=32 seats, counting each student once for every class they take.

10+13+9=32
2STEP 2

Find the extra seats

Each of the 20 students fills one seat for free, so the extra seats are 32-20=12.

32-20=12
3STEP 3

Name the all-three count

Let c be the triple-takers; then 9-c take exactly two, and the extra seats give (9-c)+2c=12.

(9-c)+2c=12
4STEP 4

Solve for the count

Combine like terms: 9+c=12, so c=3 — three students take all three classes, choice (C).

9+c=12 → c=3 → (C)
Answer
3
Check the full split: c=3 take all three, 9-3=6 take exactly two, and 20-9=11 take exactly one. Their seats add to 11· 1 + 6· 2 + 3· 3 = 11+12+9=32, matching 10+13+9. The 'at least two' count is 6+3=9, as given. The answer 3 sits inside the choices 1 through 5, so everything is consistent.
💡Key takeaway

Add every class spot (32), subtract one free spot per student (20) to get 12 repeat spots; the 9 multi-class students use 9 of them, and the 3 left over are the students taking all three — choice (C).

  • Count total class seats
  • Find the extra seats
  • Name the all-three count
  • Solve for the count

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