If one thing can be done in n1 ways, and a second thing, given the first, can be done in n2 ways, and so on for k things, then the number of ways the k things can be done is n1 x n2 x n3 ... x nk.
For example, suppose a portfolio manager is making two decisions:
Note that the multiplication rule is applicable if there are two or more groupings. In the preceding example, there are two groups: one for investment instruments and the other for countries. In addition, only one item can be selected from each group.
Suppose that there are n numbers in a group and n slots available. Only one member can be assigned to each slot. The number of ways to assign every number to the n slots is n factorial: n! = n x (n - 1) x (n - 2) x (n - 3) ... x 1. Note that by convention 0! = 1.
For example, five equity analysts are assigned to cover five industries. The number of ways to assign them is 5! = 5 x 4 x 3 x 2 x 1 = 120.
Unlike the multiplication rule, factorial involves only a single group. It involves arranging items within a group, and the order of the arrangement does matter. The arrangement of ABCDE is different from the arrangement of ACBDE.
A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula):
For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45.
An ordered listing is known as a permutation, and the formula that counts the number of permutations is known as the permutation formula. The number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does matter, is:
For example, if you select two of the ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10P2 = 10!/(10 - 2)! = 90
Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.