Problem:
A rising number, such as , is a positive integer each digit of which is larger than each of the digits to its left. There are five-digit rising numbers. When these numbers are arranged from smallest to largest, the number in the list does not contain the digit
Answer Choices:
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Solution:
The number of five-digit rising numbers that begin with 1 is , since the rightmost four digits must be chosen from the eight-member set , and, once they are chosen, they can be arranged in increasing order in just one way. Similarly, the next integers in the list begin with . So the integer in the list is the among those that begin with . Among those that begin with , there are that begin with and that begin with . Therefore, the is the of those that begin with . The first six of those beginning with are , and the seventh is . The digit is not used in the representation.
As above, note that there are integers in the list starting with either or , so the one is ninth from the end. Count backwards: . Thus is a missing digit.