Problem 2 · 2002 AMC 8
Easy
Counting & Probability
careful-counting
How many different combinations of $5 bills and $2 bills can be used to make a total of $17? Order does not matter.
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Answer: A — 2.
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Hint 1 of 2
$17 is *odd*, but stacking $2 bills can only ever build an even amount. Where does the odd part have to come from?
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Hint 2 of 2
That's a **parity** lock: the $5 bills must supply the oddness, so their count must be odd. Now you only have to try odd numbers of fives.
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Approach: use parity to pin down the number of $5 bills
- Start by noticing the odd/even clash: any pile of $2 bills totals an even number, yet the target $17 is odd. The only odd ingredient is the $5 bill, so the number of fives must be odd.
- Odd counts of fives that fit under $17: one ($5, leaving $12 = six twos) or three ($15, leaving $2 = one two). Five fives would already be $25 — too much.
- That's exactly 2 combinations.
- *Why this transfers:* whenever a total's parity doesn't match one of your building blocks, the *other* block's count is forced odd or even — parity slashes the cases before you start listing.
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