1. ## рассчитать or посчитать

- How many students they are in the bus.
- I have counted (I was pointing with my fingers on every student) that in the bus there are 25 students
- No they should be 27
- Yes I forgot to count the 2 that they returned back with private vehicles.

2. посчитать is about making basic calculations, like you said putting a finger on each object to be counted
рассчитать is about complex calculations, like when you need to first calculate a bunch of variables and then make a final calculation using those numbers. E.g. Рассчитать скорость вращения планет, рассчитать расстояние etc.

3. Я считал, пересчитал, посчитал учеников, и в итоге насчитал 25 (у меня получилось 25).

Подробнее здесь:

Кстати, цыплят по осени считают!

4. Посчитать = to count (number of students, apples, whatever)

Подсчитать = to count but using some effort (867 roubles + 300 roubles for delivery = ...)

Рассчитать = to calculate using mathematics, computer programms - the capacity in construction and so on
or (Рассчитать прислугу (old\rarely) = to pay wage to servants)

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