SAT Math: Counting Problems
Here is a question that appeared at the end of the grid-ins section of the May 2006 SAT examination.
18. In the integer 3,589 the digits are all different and increase from left to right. How many integers between 4,000 and 5,000 have digits that are all different and increase from left to right?
When confronted with this problem, many students will decide they don’t know how to approach it mathematically and, as a result, skip the problem altogether. However, there is really no reason not to attempt this problem; because it’s a grid in, you won’t be penalized for guessing the wrong answer. Although knowledge of combinations might help you solve this problem, it can also be solved by simply listing the possibilities in an organized manner.
Since the integers must be between 4,000 and 5,000, the first digit of the number must be 4. Because the integer must increase from left to right, the remaining digits must be either 5, 6, 7, 8, or 9. List possibilities one by one, starting with the smallest possibility, and moving to the largest.
4567 4568 4569
4578 4579
4589
4678 4679
4689
4789
As you can see, there are 10 integers altogether.
It is very important to be organized and systematic when listing possibilities for a problem like this. Otherwise, you are apt to miss one or two. Begin by listing all numbers that start with 456, then that start with 457, then 458, then move on to all that start with 46, and so on. Work slowly and meticulously; rushing will likewise lead to errors.
Although this problem can be solved mathematically, it would involve more advanced permutations than high school students are typically expected to know. The test-makers do not intend for you to solve this mathematically, but for all interested parties, the mathematical solution would be (5!)/(3!)(5-3!).
