If all the letters in the word 'REGULATORY' are arranged according to English alphabetical order, the positions of how many letters will remain unchanged after rearrangement?

Let's compare the original positions of the letters with their positions when sorted alphabetically:
* Original: $\text{R, E, G, U, L, A, T, O, R, Y}$
* Alphabetical: $\text{A, E, G, L, O, R, R, T, U, Y}$

Comparing term by term:

1. $\text{R}\to \text{A}$ (Changed)
2. $\text{E}\to \text{E}$ (Unchanged)
3. $\text{G}\to \text{G}$ (Unchanged)
4. $\text{U}\to \text{L}$ (Changed)
5. $\text{L}\to \text{O}$ (Changed)
6. $\text{A}\to \text{R}$ (Changed)
7. $\text{T}\to \text{R}$ (Changed)
8. $\text{O}\to \text{T}$ (Changed)
9. $\text{R}\to \text{U}$ (Changed)
10. $\text{Y}\to \text{Y}$ (Unchanged)

The positions of exactly $3$ letters ($\text{E}$, $\text{G}$, and $\text{Y}$) remain completely unchanged.