“I need someone I can count on.”
You want to hire a babysitter for you children. You're interviewing someone and you say this to explain the kind of person you want to hire.
I need someone I can count on.
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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.
When you want to describe a certain type of person, you can do it like this:
I need someone strong to help me move it.
In that example, you're talking about a "strong" person. But another way to describe a certain type of person is with a sentence, like this:
I only want to work with someone I can trust.
In this case, "I can trust them" describes the kind of person that you're talking about.
You can also include "that", "who", or "whom":
I only want to work with someone that I can trust.