The letters C, O, U, N and T are randomly placed around a circle. One such arrangement is shown here. If we always start with the C and continue to read the rest of the letters in order clockwise, in how many different orders can the letters appear?

[asy]
draw(circle((0,0),1),dashed);
label("U",dir(90));
label("C",dir(90+360/5));
label("O",dir(90+2*360/5));
label("N",dir(90+3*360/5));
label("T",dir(90+4*360/5));
[/asy]
Since we always read the letters clockwise, this is really the same as counting the number of linear permutations of the 5 letters, given that C must go first. So, there are $1 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 4! = \boxed{24}$ orders.