M35-01

Official Solution:

How many positive four-digit integers have their digits in ascending order from thousands to units?

A. \(6\)
B. \(7\)
C. \(84\)
D. \(126\)
E. \(210\)


First, note that 0 cannot be a digit in these numbers: 0 cannot be the first digit, as this would result in a three-digit number instead of a four-digit one; moreover, it cannot be any other digit, since the digits must be in ascending order and 0, being the smallest digit, cannot follow any other number.

Consequently, such numbers can only have 4 distinct digits out of the 9 possible options (excluding 0). The number of ways to choose 4 different digits from a set of 9 is \({9C4}=126\). Since there is only one ascending arrangement for each of these groups of 4 digits, 126 is the final answer. For instance, if we choose the group {3, 5, 7, 1}, the only way to arrange these digits in ascending order is 1357.


Answer: D