In this explainer, we will learn how to use the properties of permutations to simplify expressions and solve equations.

A permutation is a rearrangement of a collection of items. For example, say we have the letters A, B, and C. We can arrange them as ABC, BCA, BAC, and so forth. Each different arrangement is an example of a permutation. Notice that, for permutations,

- order matters: BCA is not the same as BAC;
- we do not allow repetition: AAB is not a valid permutation of ABC.

That is to say that a permutation represents counting without replacement in which order matters.

If we have a set of items, the total number of permutations is given by . Using factorial notation, we can write this as .

We would like to generalize this idea to counting the number of permutations of length taken from a set of , which is equivalent to the question of how many ways we can order elements from a set of elements with no repetition. We will denote this number .

We begin by considering the number of options we have for each position. For the first position, we have choices, then for the second position, we have because we have one less item to choose from. Similarly, for the third position, we have a choice of . We can continue this pattern until we have a sequence of length .

First Position | Second Position | Third Position | th Position | th Position | |
---|---|---|---|---|---|

choices | choices | choices | choices | choices |

Applying the fundamental counting principle, the total number of distinct ways we can order elements from a set of is given by the product

If we multiply this by , we can rewrite it as

Repeatedly applying the property of factorials that , we can simplify this expression to

This derivation leads us to the general definition.

### Definition: Permutations

The number of ways we can order elements from a set of elements with no repetition is given by (read as --) which is defined as The permutations are sometimes referred to as -permutations of , variations, partial permutations, or sequences without repetition. Various forms of notation are used for , the most common of which are , , and .

Notice that when , we have

Recall that, by definition, the factorial of zero is 1. Hence, , which is exactly what we expect for the number of distinct permutations of items.

In this explainer, we will develop fluency in working with this definition and learn to solve equations including . We will start with a simple example where we use the formula to evaluate .

### Example 1: Evaluating Permutations

Evaluate .

### Answer

By definition,

In this case, we have and . Therefore,

Expanding the factorial using the property that , we can rewrite this as

Canceling the from the numerator and denominator gives

Most scientific calculators have a function for evaluating permutations. So when asked to evaluate a permutation, oftentimes we can simply use a calculator to find the value. However, this is not always the case. Permutations get large very quickly so some calculators may not be able to handle numbers that large. For this reason, learning to use and manipulate the formula for permutations will be of great use. The next example will demonstrate a case where most calculators will fail to evaluate the numbers. However, by using the formula and the properties of factorials, we can evaluate the expression without the need for a calculator.

### Example 2: Ratios of Permutations

Evaluate .

### Answer

Recall the definition of permutations:

Using this definition, we can write

From here, we have two options. We could either substitute the two equations into the given expression and simplify or we could use the property of the factorial that to rewrite

Then, we can substitute this into the expression to get

In the previous example, we found that . This is actually an example of a general property of permutations that

Using this property of permutations, we can simplify expressions as the next example will demonstrate.

### Example 3: Using the Properties of Permutations to Solve Equations

If , find .

### Answer

Using the property of permutations that we can rewrite

Substituting this back into the equation, we have Canceling from both sides, we are left with as required.

We now turn our attention to a class of problems where we are given the value of in , but the unknown is in the position. Through looking at a few examples, we will highlight the techniques required to solve this class of problems.

### Example 4: Finding Unknowns in the 𝑟 Position

Find the value of such that .

### Answer

Remember that the number of permutations, , of size taken from a set of size is given by

Hence, we can write

Expanding by successively using the property that , we can rewrite this as

Canceling the common factor of , we have

Notice that this is a product of consecutive integers starting at 17 and progressively getting smaller. Hence, we would like to express 4,080 as a product of decreasing consecutive integers starting at 17. We can do this by dividing 4,080 by 17, then by 16, then by each successive smaller integer. Starting with 17, we have

Hence,

We now divide 240 by 16 which gives 15. Hence, which is a product of successive integers. Therefore, we have expressed as a product of consecutive integers as

Comparing this with equation (1), we can equate the smallest integers to get . By rearranging, we find that .

We can double check that we have the correct answer by using a calculator to confirm that .

### Example 5: Solving Permutation Equations for Unknowns in the 𝑟 Position

Solve the following equation for :

### Answer

Recall the definition of permutations: Using this, we can write

Using the property that , we can expand to get

Canceling the common factor of , we have that

which is a product of consecutive decreasing integers starting with 5. We would, therefore, like to express 120 in the same form. We can do this by dividing 120 by 5, then 4, then each consecutive smaller integer as follows:

Hence, we have

If we compare this with equation (2) and try to equate the smallest integers, we find that the smallest integer could be either 2 or 1. Hence, we have two possible cases, when , or when . Rearranging, we get that or .

We can double check both of these values by using a calculator to confirm that and .

In the next couple of examples, we use the definition of and the properties of factorials to solve permutation problems by simplifying the equations to a linear or quadratic form.

### Example 6: Solving Permutation Equations

If , find the value of .

### Answer

Recall that . Using this, we can write

Substituting this into the given equation, we get

Canceling common factors gives us

Using the property of factorials that , we can rewrite

Substituting this back into our equation gives

Dividing both sides of the equation by yields

Rearranging, we get the following quadratic equation:

Using the quadratic formula or factoring, we can solve this to find that or . Since factorials and permutations are only defined for positive integers, we disregard the solution. Thus, .

### Example 7: Solving Permutation Equations

Find the value of given that .

### Answer

Using the definition of , we rewrite the given equation as

Dividing by , we can rewrite this as

Multiplying by , we get

Using the fact that , we rewrite this as

Canceling the common of from the numerator and denominator gives

By gathering like terms and rearranging, we get

Dividing by 4, we have .

In the final example, we look at the case where we are given the value of in , but the unknown is in the position. This case is less obvious as to how to solve it. However, we will use an example to show a technique that can be applied in cases like this.

### Example 8: Solving Permutation Equations for Unknowns in the 𝑛 Position

Find the value of such that .

### Answer

Remember that the number of permutations, , of size taken from a set of size is given by

Therefore, using the property of factorials that , we can write

Therefore, we are looking for three consecutive integers whose product equals 32,736. There are two ways to do this, for smaller numbers, we can consider the prime factors of the number and then rearrange them to form three consecutive numbers. This can work for larger numbers too, as we will show. However, in general, this process could prove increasingly difficult for larger and larger numbers. However, there is a useful trick we can use: taking the cube root of 32,736 gives us a number that when multiplied by itself twice gives 32,736. Therefore, the value of this number will be close to the value of the three consecutive numbers we are looking for.

For 32,736, we have

We therefore divide 32,736 by the two integers on either side of this value: 31 and 32. When we do this we find which is the product of three successive integers as required. Hence, we see that .

As mentioned above, we can consider the prime factors of 32,736 and try to rearrange them into three consecutive integers. For 32,736, we find that one of the prime factors is 31; hence, we are able to easily organize the other factors into two other consecutive integers as the calculation below demonstrates: from which we can conclude .

### Key Points

- The number of permutations of size taken from a set of size is given by
- Some of the key techniques to help solve problems involving
are as follows:
- Using the property of factorials that
- Using the property of permutations that
- We often need to divide a number by consecutive integers. If we are given in , we often divide by consecutive decreasing integers starting from . However, if we are given in , we know the number of consecutive integers but not where to start. Therefore, we can either consider prime factors or use the integers on either side of the th root as a starting point.