Let=Hire=Rent:
You rent tools, house, fixtures etc: I am going to rent a car, driller, computer, house etc
You hire people, contractors: I need to renovate my house, I am going to hire a contractor
Let? I dont know how it is related to the other two!
The number of=A number of
The number of: use it when you know the number: The number of students who participated in this discussion was 121/high/ a lot
A number of : use it when you dont know the figure and have no clue/ or you dont care about the number:
A number of students participated in this discussion ( means you dont care how many, you just wanna say there were more than one/ it also means you dont know how many )
In general: The "reads THAA" means you know what you and your audience both know what is being referred too.
Jack:I wanna tell you the joke I told john, you wanna here it? ( means Jack is going to tell you about a joke that you heard some story about. You know this joke was told to John.
I wanna tell you a joke, would you like to hear it? means there is a random joke that is bout to be told to you.