You want to hire a babysitter for you children. You're interviewing someone and explaining the kind of person you want to hire. You say:
I need someone I can count on.
To "count on" someone means to trust them to do a job. For example, at work you can ask an employee:
Can I count on you to have this done by Monday?
When you "trust" someone, it means that you believe that they will tell you the truth and not do things to hurt you. But when you "count on" a person, you believe that they'll be able to do a job correctly. So it's possible to trust a person but not necessarily be able to count on them.
When you want to say what the job that you trust the person to do is, use "count on ___ to ___":
I'm counting on you to tell me if I make any mistakes.
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