Problem 13 · 2024 Math Kangaroo
Stretch
Counting & Probability
careful-countingcasework
The number 2024 is made up of the four digits 2, 0, 2 and 4. How many different four-digit numbers bigger than 2024 can be formed using exactly those digits?
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Answer: E — 8
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Hint 1 of 2
Only the leading digit can be 2 or 4 (a leading 0 is not a four-digit number); split into numbers starting with 2 and numbers starting with 4.
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Hint 2 of 2
List the rearrangements in each case and keep just those larger than 2024.
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Approach: split by leading digit and count those above 2024
- Using the digits 2, 0, 2, 4, the first digit must be 2 or 4.
- Starting with 2, the last three digits arrange the set {0, 2, 4}: 2024, 2042, 2204, 2240, 2402, 2420, of which five exceed 2024.
- Starting with 4, every arrangement beats 2024: 4022, 4202, 4220, giving three more.
- Total: 5 + 3 = 8 numbers bigger than 2024.
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