32. Find the number of 11 -digit positive integers such that the digits from left to right are non-decreasing. (For example, $12345678999,55555555555,23345557889$.)
32. Answer: 75582
We note that each 19-digit binary sequence containing exactly eight ' 0 's and eleven ' 1 's can be matched uniquely to such an 11-digit positive integer in the following way: Each ' 1 ' will be replaced by a digit from 1 to 9 , and one more than the number of ' 0 's to the left of a particular ' 1 ' indicates the value of the digit. For example, 0110000010101111111 is matched to 22789999999 , and 1110000000011111111 is matched to 11199999999 . It follows that the required number is $\binom{19}{8}=75582$