Level 1 CFA® Exam:
Principles of Counting
In this lesson, we're going to briefly discuss the fundamental principles of counting to be used in your level 1 CFA exam. Here are some problems that you'll be able to solve after reading this lesson:
- how many three-digit numbers can be constructed in which the first digit can only be 1, 2, or 3, the second can be 4, 5, or 6, and the third – 7, 8, or 9,
- how many ways there are to arrange six books on a shelf,
- how many possible winning lottery results there are if 6 numbers are drawn out of 49,
- how many possible ways there are of forming a delegation of three from among a team of 10 members if we assume that the order in which the members are drawn is important.
The principles of counting can also be useful for determining probability. For example, we can calculate how probable it is that we draw the number 147 or 258 or 369 if we know that the first digit must be 1, 2, or 3, the second must be 4, 5, or 6, and the third must be 7, 8, or 9.
We can also calculate the probability that the books are lined up on a shelf from volume 1 to volume 6.
We can also find out about the probability of winning a lottery in which 6 numbers are drawn out of 49 if we buy one coupon or the probability that a 3-person delegation of a 10-member team includes 3 youngest members.
The general principle for computing probability is that we divide outcomes favorable for what we're examining by the entire sample space (or all the possible outcomes). We can do that assuming that all possible outcomes are equally likely.
You have to know that concepts related to the principles of counting which we're discussing here are often used in finance, for example in the binomial option pricing model or problems related to the timing of investments.
The multiplication rule of counting is appropriate if the outcome of a task depends on a sequence of decisions.
For example, when making the first decision we have a choice of \(n_1\) options, when making the second decision we have \(n_2\) options and so up to \(n_k\). This is how we know there are:
\(n_1\times{n_2}\times{n_3}\times{…}\times{n_k}\)
ways to complete the task.
- How many three-digit numbers can be constructed in which the first digit can be 1, 2, or 3, the second can be 4, 5, or 6, and the third is 7, 8, or 9?
- What's the probability that we draw a number 147, 258, or 369 from among all the possible numbers?
(...)
The factorial of a number 'n' is the product of all natural numbers less than or equal to 'n'. The factorial of a number 'n' ('n' factorial) is denoted by \(n!\).
For example:
- \(3!=1\times2\times3=6\)
- \(5!=1\times2\times3\times4\times5=120\)
Computing factorials is easier using a financial calculator. In the Texas Instruments Professional calculator, the factorial is labeled as \(x!\) and it's the second function of the multiplication key, so if you want to find \(5!\), you need to press the following sequence of keys:
5 [2nd] [\(\times\)] and the result is 120
'n' factorial allows us to compute the total number of possible n-element sequences comprising all the elements of a set of 'n' elements.
- How many ways there are to arrange 6 books on a shelf?
- What is the probability that the books will be arranged in ascending order from volume 1 to volume 6?
If we want to arrange 6 books on a shelf, we can do it in:
\(6!=1\times2\times3\times4\times5\times6=720\text{ ways}\)
We can deal with that in the following way: the first position on the shelf can be taken by one of six books, the second – by one of the five remaining books, the third – by one of the four remaining books, and so on. As you can see, there are:
\(6\times5\times4\times3\times2\times1=\\=720\text{ possible arrangements}\)
The probability that the books will be arranged in ascending order from volume 1 to volume 6 therefore equals to:
\(P(\text{ascending})=\frac{1}{720}\)
We use the combination formula when we want to choose 'r' objects from an 'n'-element set.
Obviously, 'r' must be lower than 'n'. The number of combinations can be calculated using this formula:
Note that in the case of the combination formula, the order in which the elements in the subsets are listed doesn't matter, so the result doesn't depend on the arrangement of elements in the subsets.
The combination formula can be used to compute all the possible results of a lottery in which 6 numbers are drawn from a set of 49 numbers.
- How many possible results are there?
- What is the probability of choosing the right six numbers?
(...)
As mentioned above in the case of the combination formula the order of elements in the set does not matter. And to win most lotteries it's enough to get the right numbers and they don't have to be arranged in the order in which they are drawn.
If we want to know the number of possible 'r'-element sub-sets of an 'n'-element set in which the order of elements matters, we need to use the permutation formula. A permutation is an ordered 'r'-element sub-set of an 'n'-element set. When we say 'ordered' we mean that the order of elements matters. The permutation formula looks as follows:
We want to form a 3-member delegation out of a team of 10. We need each delegate to serve a different function. One person will be responsible for presenting a new, innovative product; another one – for talks with potential customers; and the third person will deal with organizational matters.
- How many ways there are of forming the delegation?
- What is the probability that the delegation includes the 3 youngest members of the team of 10 people?
(...)
- The multiplication rule of counting is appropriate if the outcome of a task depends on a sequence of decisions.
- The factorial of a number 'n' is the product of all natural numbers less than or equal to 'n'. The factorial of a number 'n' ('n' factorial) is denoted by \(n!\).
- We use the combination formula when we want to choose 'r' objects from an 'n'-element set.
- If we want to know the number of possible 'r'-element sub-sets of an 'n'-element set in which the order of elements matters, we need to use the permutation formula.