## 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

SOLUTION
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.
$481\to 486\!:512346,\cdots,512643$
$487\to 492\!:513246,\cdots,513642$
$493\to 498\!:514236,\cdots,514632$
$499\!:516234$
$500\!:516243$

Answer: $516243$