## Monday, July 23, 2007

### How many even 3-digit positive integers do not contain the digits 2,4, or 6?

More practice for students...
As indicated many many times on this blog, there is no substitute for experience!
The keys to success here are:
(1) Careful reading (do students often miss the key word!)
(2) Knowledge of facts (why is zero the most important number in life!)
(3) Knowledge of strategies, methods (multiplication principle, organized lists, counting by groups, etc)

If these problems are helpful for students, let me know...

novemberfive said...

We want to find the number of even integers x such that 100 <= x <= 999. In order to ensure that 100 <= x, the first digit must be nonzero. The remaining choices for the first digit are {1,3,5,7,8,9}. The only restriction on the second digit is that it cannot be 2,4, or 6, so it must be one of {0,1,3,5,7,8,9}. To make the number even, the last digit must be even; indeed, it must be either 0 and 8.

There are 6 ways to choose the first digit, 7 ways to choose the second digit, and 2 ways to choose the third digit. Thus, the answer is 6 * 7 * 2 = 84.

Mike said...

Let's see ... I'm figuring that 0 cannot be used in the hundreds place, since that would allow for a number such as 038, which I would say is not a 3-digit integer. So, in the hundreds place, the possibilities are 1,3,5,7,8,9. In the tens place, the possibilities are 0,1,3,5,7,8,9. In the ones place, the possibilities are 0,8. So, multiplying the number of possibilities together, we get 6 x 7 x 2 = 84 even 3-digit positive integers that do not contain the digits 2,4, or 6 ...

Dave Marain said...

novemberfive and mike--
excellent explanations!
I encourage both students and educators to offer solutions, comments, suggestions.
I would be interested in your thoughts about the appropriateness of these kinds of questions for the classroom. These questions do not fit naturally into a traditional algebra or geometry curriculum. Some algebra textbooks do include the multiplication principle as part of probability. Many students see few of these until later on in math. Where would you include it? OR do you see these as just SAT-prep to be included if time permits...

jonathan said...

You know that I use this sort of question for my "off topic problem solving to develop problem solving skills," right?

See how much I miss by taking a trip!

Since we value alternate solutions...

900 3 digit numbers. Half (450) are even. Two-thirds (300) don't start with 2,4, or 6. 40% (120) end with 0 or 8. And 70% (84) don't have an illegal middle digit.