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.
This entry was posted in Problem solving and tagged , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s