I understand you are taking a college course in mathematics and studying permutations and combinations. Permutations and Combinations date back through the ages. According to Thomas & Pirnot (2014), there is evidence of these mathematical concepts as early as AD 200. As we solve some problems you will see why understanding these concepts is important especially when dealing with large values.
I also understand you are having problems understanding their subtle differences, corresponding formulas nPr and nCr and the fundamental counting principle. Before we review some exercises, I would like to provide you with some definitions you will need in solving some problems.
According to Thomas & Pirnot (2014), a permutation is an ordering of distinct objects in a straight line. If we select r different objects from a set of n objects and arrange them in a straight line, this is called a permutation of n objects taken r at a time. The number of permutations of n objects taken r at a time is denoted by P (n,r).
In working with permutations and combinations we are choosing r different objects from a set of n objects. The big difference is whether the order of the objects is important. If the order of the objects matter, we are dealing with permutations. If the order does not matter, then we are working with combinations (Thomas & Pirnot, (2014), p. 624, 626).
According to Thomas & Pirnot (2014), in combinations if we are choosing r objects from a set of n objects and are not interested in the order of the objects, then to count the number of choices, we must divide P (n,r)* by r!. We now state this formally: Formula for computing C(n,r). If we chose r objects from a set of n objects we say we are forming a combination of n objects taken r at a time. The notation C (n,r) denotes of number of such combinations (Thomas & Pirnot, (2014), p. 626).
Thomas & Pirnot (2014) explanation of The Fundamental Counting Principle (FCP) is as follows: If we want to perform a series of tasks and the first task can be done in a ways, the second can be done in b ways, the third can be done in c ways, and so on, then all the tasks can be done in a x b x c x…ways (Thomas & Pirnot, (2014), p. 616).
Eight books are placed on a shelf. How many ways are there to arrange...