18. Find the number of 7-digit positive integers such that the digits from left to right are nonincreasing. (Examples of 7-digit non-increasing numbers are 9998766 and 5555555; An example of a number that is NOT non-increasing is 7776556 )
18. Answer: 11439
Each 16 -digit binary sequence containing exactly nine ' 0 's and seven ' 1 's can be matched uniquely to such a 7 -digit integer or 0000000 as follows: Each ' 1 ' will be replaced by a digit from 0 to 9 in this way: the number of ' 0 's to the right of a particular ' 1 ' indicates the value of the digit. For example, $0110000010101101 \sim 8832110$ and $1111000000000111 \sim 9999000$. Thus, required number $=\binom{16}{7}-1$.