How many counting numbers have four distinct nonzero digits such that the sum of the four digits is 11?

To find how many counting numbers have four distinct nonzero digits that sum up to 11, we first identify all combinations of four distinct digits (from 1 to 9) that meet this criterion. The possible combinations of digits that add up to 11 are limited, and we can use combinatorial methods to list them. After identifying valid sets, we can calculate the permutations for each set (since the order of digits matters) to get the total count. Upon calculating, we find there are 24 valid combinations yielding 576 distinct numbers.