To "count on" someone doing something means that you totally trust that they'll do it. You don't prepare for the possibility that they might not do it.

For example, if a person has spent all her money and doesn't have enough to pay for her rent on the 30th of the month, you could say:

She's counting on getting paid on the 25th.

That means that she strongly believes she'll get paid and doesn't have a plan for what to do if the payment doesn't come.

When the thing you're planning on is a person's action, you say "count on (someone)(doing something)":

Don't count on Sandra being available on Friday. She'll probably be busy then.