1. What is the smallest four-digit positive integer which has four different digits?
A 1032
В 2012
C 1021
D 1234
E 1023
Solution: $\mathbf{E}$
Here it is easy just to check the options that are given. A, D and E are the only options in which all four digits are different. Of these, clearly, $\mathrm{E}$ is the smallest.

For a complete solution we need to give an argument to show that 1023 really is the smallest fourdigit positive integer with four different digits. It is easy to do this.
To get the smallest possible number we must use the four smallest digits, $0,1,2$ and 3 . A four digit number cannot begin with a 0 . So we must put the next smallest digit, 1 , in the thousands place, as a four-digit number beginning with 2 or 3 is larger than one beginning with a 1 . For similar reason the hundreds digit must be the smallest remaining digit, 0 . Similarly the tens digit must be 2 and the units digit must be 3 . So the required number is 1023 .