Counting problems, permutations, and combinations

This section has a lot in common with probability. Combination problems can be solved using the methods we learned with probabilities. Permutations, however, require that we use memorized formulas for combinations and permutations.

Combinations

If probabilities tell us how likely something is to happen, combinations tell us how many possible outcomes there are.

When solving a probability problem, we take all of the relevant probabilities and multiply them together as fractions. The result is a smaller fraction, showing that the probability of an event occurring decreases as it becomes more specific.

Combination problems are treated similarly, but instead of treating them as fractions, we treat them as whole numbers. If there are two possibilities, the corresponding probability would be $1/2$, while the corresponding combination would be $2$. Let’s walk through an easy example first and then a more difficult one:

There are $6$ cars and $4$ drivers. How many possible combinations are there of drivers in their respective cars?

If this were a probability question, we would say that there is a $1/6$ probability of picking a certain car and a $1/4$ probability of picking the driver who would drive it. So, for combinations, we look at the reciprocal. There are $6$ options for cars and $4$ options for drivers. Multiply those numbers together to get the total number of combinations: $24$.

Permutations

Permutations are more complex than combinations. If combinations describe how many possible outcomes there are, permutations tell us how many patterns there are between all the different combinations. In other words, for combinations, the order of the items does not matter. For permutations, the order of the items in the configuration does indeed matter.

The formula to determine a permutation ($P$) is:

$$P=\frac{n!}{(n-r)!}$$

where $n$ is the number of objects in the problem and $r$ is the number of things we take from the total.

This formula introduces a new mathematical symbol ($!$) called a factorial. This symbol indicates that the number it is attached to must be multiplied by every whole number that precedes it, decreasing to $1$. So, $5!=5\ast 4\ast 3\ast 2\ast 1=120$.

You can either write out the value of a factorial by hand, or you can locate the factorial symbol on your calculator. For the TI-84, the factorial symbol is found under the “math” button and the “probability” tab within it. We will do an example to help make sense of this:

There is a table with four seats and five possible guests to fill it. How many possible combinations of seat arrangements are there?

Let’s first solve this problem with the formula, then understand the math behind it.

Remember our formula? $P=\frac{n!}{(n-r)!}$

For this question, $n=5$ and $r=4$.

$$\begin{array}{rl}P& =\frac{5!}{(5-4)!}\\ & \\ P& =120\end{array}$$

The first seat has five options of who sits in it. After the first person sits down, there are four options for the next seat. The third seat only has three options of who sits in it, and the last person has two options. The number of possibilities, therefore, consists of $5$, $4$, $3$, and $2$. Multiplying these numbers together, we get the total number of permutations, or combinations that depend on order: $120$.