10. Principles of Counting

o. identify the most appropriate method to solve a particular counting problem and solve counting problems using the factorial, combination, and permutation concepts.

If you have 5 equity analysts assigned to cover 5 territories, what is the total number of ways all 5 can be assigned to a territory? - Use the factorial rule for this kind of “multiple slot” problem - n! = 5 * 4 * 4 * 2 * 1 = 120

If you have 5 equity analysts and 3 territories, what is the total number of ways that one of them can be assigned to one territory? - Use the multiplication rule when only one item can be chosen from each set - \(n = 5 * 3 = 15\)

What is the combination formula (binomial formula)? This is used to choose n slots from an unordered group: - \(_n C_r = \frac{n!}{(n-r)! \times r!}\) - n = total number of objects - C_r = combination of r number of objects - If you have five analysts and you want to pick 3 of them: - 5! / (5 - 3)! * 3! = 10

What is the permutation formula and what is the mathematical formula? - Permutation formula tells us the total 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 - \(_n P_r = \frac{n!}{(n - r)!}\)

Can there ever be more combinations than permutations for the same problem? Why? No, there can’t be more combinations than permutations because permutations take into account all possible orderings of items and combinations do not

A portfolio has 10 stocks. Each stock can be sold or kept in the portfolio. What is the possible number of outcomes? - \(2^{10} = 1,024\) - 2 possible outcomes, 10 stocks