How many distinct odd-digit numbers can be written with the digits 1,2,3, and 4 if no digit may be used more than once?

I assume you meant odd numbers rather than odd-digit numbers.

 

For a number to be odd, it has to end with an odd digit; in your case, it has to end with either 1 or 3.

 

If the number ends with a 1, that leaves 2, 3 and 4 to "fill up" the first three spots of a four-digit number.

2, 3, and 4 can be arranged in 6 distinct ways, meaning there are 6 numbers that fit your criteria when the number ends with 1.

 

The same applies when the number ends with 3, for which exist another 6 distinct numbers that fit your criteria.

 

6+6 = 12 total numbers that fit your criteria.