Problem 8 · 2004 AMC 8
Easy
Counting & Probability
constrained-counting
Find the number of two-digit positive integers whose digits total 7.
Show answer
Answer: B — 7.
Show hints
Hint 1 of 2
You don't need to track two digits at once. Once you pick the tens digit, the units digit is forced (it has to finish the sum to 7). So the only real question is: how many legal tens digits are there?
Still stuck? Show hint 2 →
Hint 2 of 2
The principle is one free choice fixes the rest: count only the variable you're actually free to choose. Here a two-digit number can't start with 0, so the tens digit runs 1 to 7 — that's the entire count.
Show solution
Approach: count only the free digit
- Pick the tens digit a; the units digit is then forced to 7 − a. So the count equals the number of valid a.
- a can be 1, 2, 3, 4, 5, 6, or 7 — it can't be 0 (no leading zero) and can't exceed 7 (then the units digit would go negative). That's 7 values: 16, 25, 34, 43, 52, 61, 70.
- You'll reuse this whenever one choice locks in the others — count the driver, not the whole pair.
Mark:
· log in to save