Problem 28 · 2014 Math Kangaroo
Stretch
Counting & Probability
careful-countingplace-value
Consider all 7-digit numbers that use each of the digits 1 to 7 exactly once. Write these numbers in increasing order and split the list exactly in the middle into two lists of equal size. What is the last number of the first list?
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Answer: E — 4376521
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Hint 1 of 2
There are 7! such numbers in order; the split point is right in the middle.
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Hint 2 of 2
Find the 2520th smallest arrangement of the digits 1–7.
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Approach: locate the middle permutation by digit blocks
- There are 7! = 5040 numbers, so the first list ends at the 2520th smallest.
- Counting in blocks (each leading digit gives 6! = 720 numbers): 2520 = 3·720 + 360, landing among the numbers starting with 4, then resolving the rest.
- The 2520th number is 4376521.
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