Six-digit Numbers

If all the six-digit numbers formed by using the digits 1,2,3,4,5, and 6, without repetition, are listed from least to greatest, which number will be 500^{th} in the list?
Source: NCTM Math Teacher 2008

There are 6\times 5\times 4\times 3\times 2\times 1=720 six-digit numbers.
1\to 120\!:123456,\cdots,165432
121\to 240\!:213456,\cdots,265431
241\to 360\!:312456,\cdots,365421
361\to 480\!:412356,\cdots,465321
481\to 600\!:512346,\cdots,564321
601\to 720\!:612345,\cdots,654321
We examine the 481\to 600 numbers in finer detail.
481\to 486\!:512346,\cdots,512643
487\to 492\!:513246,\cdots,513642
493\to 498\!:514236,\cdots,514632

Answer: 516243


About mvtrinh

Retired high school math teacher.
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