Problem 18 · 2016 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
Richard writes down all numbers with the following properties: the first digit is 1; each of the following digits is at least as big as the previous one; the sum of the digits is 5. How many such numbers can Richard write down?
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Answer: B — 5
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Hint 1 of 3
The first digit is 1, so the digits after it must add up to the remaining 4.
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Hint 2 of 3
Those later digits can never go down, so list them from smallest going up.
Still stuck? Show hint 3 →
Hint 3 of 3
Just write out every non-decreasing way to make 4 (using digits at least 1).
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Approach: list every non-decreasing way the leftover digits add to 4
- The number starts with 1, so the digits that follow are non-decreasing and add up to \(5 - 1 = 4\).
- Listing the ways to make 4 without ever decreasing: 4, then 1+3, then 2+2, then 1+1+2, then 1+1+1+1.
- These give the numbers 14, 113, 122, 1112, 11111 — that is 5 numbers, choice (B).
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