2006 Alabama ARML TST Problems/Problem 4
Problem
Find the number of six-digit positive integers for which the digits are in increasing order.
NOTE: Increasing from left to right. Digits are distinct.
Solution
Solution 1
Think of a nine-digit number . If you take out digits, then it will become a -digit number and all the digits will still be in increasing order. The number of ways to take three digits out is
Solution 2
We pick six different digits through for the integer. None of them can be 0, or else it is a five digit integer or the digits are not in increasing order. Let's say that is the least digit of them all. is therefore the hundred-thousands digit. Let's say that is the second smallest integer. Then is the ten-thousands digit. etc.
For each group of we pick, there is only one arrangement such that each digit is in increasing order. There are ways to pick the digits, therefore there are 84 integers.
See also
2006 Alabama ARML TST (Problems) | ||
Preceded by: Problem 3 |
Followed by: Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |