Counting Method for Determining Unique Combinations | CFA Level 1 Exam

B

The multiplication rule of counting states that the number of combinations available when there are n_1 options in one aspect, n_2 in another, and so on, up to n_k, is n_1 * n_2 * ... * n_k.

The counting method that should be used to determine the number of unique combinations in a process involving different options in each respect is the multinomial formula, which corresponds to answer choice C.

The multinomial formula is used when we want to count the number of ways to arrange items into distinct categories, where each category can have a different number of options. It is an extension of the concept of combinations and permutations.

Let's break down the other answer choices to understand why they are not appropriate in this context:

A. The binomial formula: The binomial formula is used when we have two options (e.g., success or failure) and want to determine the number of combinations or probabilities associated with different outcomes. It is not suitable for situations with more than two options.

B. The multiplication rule: The multiplication rule is used when we want to determine the number of outcomes of independent events occurring together. It involves multiplying the number of options in each event to calculate the total number of possibilities. However, in this case, the options are not independent but rather categorized into different respects, so the multiplication rule is not applicable.

D. The permutation rule: The permutation rule is used when the order of the elements matters. It calculates the number of arrangements of items when the order is important. However, in this case, the question does not specify the order of the options within each respect, but rather the total number of unique combinations. Therefore, the permutation rule is not suitable either.

The multinomial formula, on the other hand, is used to count the number of ways to distribute items into distinct categories when each category can have a different number of options. It is expressed as:

n! / (n₁! × n₂! × n₃! × ...)

where n is the total number of items, and n₁, n₂, n₃, etc., represent the number of options in each category. The factorial symbol (!) denotes the product of all positive integers up to that number.

By using the multinomial formula, we can accurately determine the number of unique combinations that can result from a process involving different options in each respect.