Rely, trust, depend, count, reckon, bank can all mean to have or place full confidence.
One relies on or upon someone or something that one believes will never fail in giving or doing what one wishes or expects. Rely usually connotes a judgment based on previous experience and, in the case of persons, actual association.
One trusts, or trusts in or to, when one is completely assured or wholly confident that another will not fail one in need.
Trust stresses unquestioning faith which need not be based on actual experience.
One depends on or upon someone or something when one, with or without previous experience, rests confidently on him or it for support or assistance.
Depend may connote a lack of self-sufficiency or even weakness; it often implies so strong a belief or so confident an assumption that the hoped-for support or assistance is forthcoming that no provision for the contrary is made.
One counts or reckons on something when one takes it into one’s calculations as certain or assured; the words often imply even more confidence in expectation than depend and may carry even more than the latter’s frequent suggestion of possible distress or disaster if one’s expectations are not fulfilled.
But these terms are often weakened in use to the point that they mean little more than expect .
One banks on something or someone in which one’s confidence is as strong as it would be in a bank to which one would entrust one’s money; hence, the term is appropriate when one wishes to express near absolute certainty without any of the other implications inherent in depend, count, and reckon .