2020 AMC 8
Complete problem set with solutions and individual problem pages
How many integers between and have four distinct digits arranged in increasing order? (For example, is one integer.).
- A.
- B.
- C.
- D.
- E.
Solution 1
Firstly, we can observe that the second digit of such a number cannot be or because the digits must be distinct and in increasing order. The second digit also cannot be as the number must be less than , so the second digit must be . It remains to choose the latter two digits, which must be distinct digits from . That can be done in ways; there is then only way to order the digits, namely in increasing order. This means the answer is .
Solution 2
As in Solution 1, we find that the first two digits must be , and the third digit must be at least because the digits can not repeat. If it is , then there are choices for the last digit, namely , , , , or . Similarly, if the third digit is , there are choices for the last digit, namely , , , and ; if , there are choices; if , there are choices; and if , there is choice. It follows that the total number of such integers is .
