LOS O requires us to:
identify the most appropriate method to solve a particular counting problem and solve counting problems using factorial, combination, and permutation concepts.
1. Multiplication Rule
a. If one task can be done in n1ways, and a second task, given the first, can be done in n2 ways, and a third task, given the first two tasks, can be done in n3 ways, and so on for k tasks, then the number of ways the k tasks can be done is [(n1)(n2)(n3) … (nk)].
b. Thus for example, if we have 4 shirts, 4 pants, 3 jackets, and 5 ties; then the number of dresses that can be made out of these are:
c. Taking another example, if we have three analysts who need to cover three different industries. Then, the first analyst can be assigned any of the three industries, the second analyst can be assigned any out of the remaining two industries, and the third analyst is left with only one option. Thus the industries can be assigned to the three analysts in (3*2*1) or 3! ways.
2. Multinomials
a. The number of ways that objects can be labeled with k different labels, with n1 of the first type, n2 of the second type, and so on, with n1 + n2 + … + nk = n is given by:
b. For example, if we have to divide 20 students into 3 groups. The first group must have 8 students, the second group must have 7 students and the third group must have 5 students, and the order does not matter, then the number of possible combinations are:
3. Combinations
a. If we have to select r objects from a group of n units, we can do so using the following formula:
b. Thus if we have to choose a combination of 3 companies out of a pool of 5 companies, then we can do so in the following number of ways:
4. Permutations
a. Now, if the order of the combination matters, and we have to choose r objects from out of n units, we can do so using the following formula:
b. Thus, in the above example, we have to choose the three companies out of 5 and rank them in the order of I, II, and III; then we can do so in the following different ways: