Problem 18 · 2012 Math Kangaroo
Hard
Counting & Probability
careful-countingsum-constraint
How many numbers from 1000 to 9999 are there which have 3 as the hundreds digit, and for which the sum of the remaining three digits is also 3?
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Answer: E — 6
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Hint 1 of 2
Fix the hundreds digit as 3; the other three digits (thousands, tens, units) must sum to 3.
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Hint 2 of 2
Count the ways with the leading digit at least 1.
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Approach: count digit choices under a sum constraint
- The hundreds digit is fixed at 3; the thousands, tens and units digits must sum to 3, with the thousands digit at least 1.
- List the (thousands, tens, units) options: (1,1,1), (1,0,2), (1,2,0), (2,0,1), (2,1,0), (3,0,0) — giving 1311, 1302, 1320, 2301, 2310, 3300.
- That is 6 such numbers (E).
Mark:
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