What is the difference between nPr and nCr in permutations and combinations?

The key difference is whether order matters. Use permutations nPr when arrangement or order is important, like seating people in a row or forming numbers. Use combinations nCr when only the selection matters and order does not, like choosing a team or a committee. The formulas are nPr = n! divided by (n − r)! and nCr = n! divided by [r!·(n − r)!]. Notice nPr = nCr × r!, because each chosen group of r items can be arranged in r! orders. A quick test: ask yourself if swapping two chosen items gives a genuinely different outcome. If yes, it is a permutation; if it is the same outcome, it is a combination. For example picking 3 winners with ranks uses nPr, picking 3 prize winners equally uses nCr.