AMC 10 · 2018 · #5

Grade 7 counting
Archetype Overlap / Complement Counting └ Sub-archetype Complement by Subtracting Forbidden Arrangements
prime-numberscombinations-basicexponents complementary-countingsystematic-enumeration ↑ Prerequisites: prime-numberscombinations-basic
📏 Medium solution 💡 2 insights
Problem
Count how many subsets of the eight-element set of the numbers 2 through 9 contain at least one prime number.

Pick an answer.

(A)
128
(B)
192
(C)
224
(D)
240
(E)
256
How to solve
Strategy Change Focus / Count the Complement

Tool #16 (Change Focus / Count the Complement): "at least one prime" is awkward to count head-on because a subset could have one, two, three, or four primes. The opposite event — "no prime at all" — is a single clean case, so we count those and subtract from the total. Tool #2 (Make a Systematic List): first sort the eight numbers into primes and composites so we know exactly how many of each there are. Tool #3 (Eliminate Possibilities): the result must be one of the five choices, which lets us confirm 240 and rule out the rest.

1STEP 1

Sort into primes and composites

Sort into primes and composites.

primes={2,3,5,7}, composites={4,6,8,9}
2STEP 2

Flip to the complement

At least one flips to the complement.

#(at least one prime) = #(all) - #(no prime)
3STEP 3

Count all subsets

All subsets number two to the eighth.

2⁸ = 256 subsets total
4STEP 4

Count subsets with no prime

Prime-free subsets use composites only.

2⁴ = 16 subsets with no prime
5STEP 5

Subtract to finish

Subtracting gives 240.

256 - 16 = 240 → (D)
Answer
240
The answer 240 is just under the total 256, which makes sense: only 16 of all subsets avoid every prime, so the vast majority must contain at least one. The split 4 primes and 4 composites is balanced, and 2⁸ - 2⁴ = 256 - 16 = 240 is exact. Choices (A) 128 and (B) 192 are far too small, (E) 256 would wrongly count every subset including the empty set, and (C) 224 = 256 - 32 would mistakenly use 2⁵ composites — only (D) fits.
💡Key takeaway

To count subsets with "at least one" prime, count all 2⁸ = 256 subsets, subtract the 2⁴ = 16 that use only composites, and you get 240.

  • Sort into primes and composites
  • Flip to the complement
  • Count all subsets
  • Count subsets with no prime
  • Subtract to finish

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