Permutations and combinations are ways of representing groups of objects by selecting them from a set and forming subsets. With permutations and combinations, we can organize certain groups of data. We will start by looking at the definitions of permutations and combinations. Then, we will learn how to use their formulas by solving some exercises.

## Definition of permutations

Permutations refer to the action of organizing all the elements of a set in some kind of order or sequence. This means that if a set is already ordered, the process of rearranging its elements is called permuting.

With permutations, the order of the elements does matter. If our password is 1234 and we enter the numbers 3241, the password will be incorrect since we have the same numbers, but in a different order. This means that 3421 is a permutation of 1234.

### EXAMPLE

The permutations of 1, 2, 3, 4 are:

- 4321, 4312, 4123, 4132, 4213, 4231, 3412, 3421, 3214, 3241, 3124, 3142, 2413, 2431, 2314, 2341, 2134, 2143, 1432, 1423, 1324, 1342, 1234, 1243.

## Definition of combinations

A permutation is related to the action of organizing the elements of a set so that, unlike permutations, the order of the selection does not matter. For example, choosing a team of 3 people from a group of 20 people is a combination.

### EXAMPLE

If we have the numbers 1, 2, 3, 4, 5 and we have to choose 3 numbers, we can obtain the following sets:

- 123, 234, 345, 124, 125, 134, 145, 135, 235, 245.

These are the only possible sets since by choosing 123, we will get the same numbers as 132, 213, 231, 321, 312 just in a different order.

## Permutations and combinations formulas

The formulas for permutations and combinations can have different variations, but the three most important are:

### Permutations formula

If we have a collection of *n* objects, then the number of ways we can choose *r* of them is equal to:

$latex _{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}$ |

### Combination formulas

If we don’t want to take into account the different permutations of the elements, we can divide the expression above by the number of permutations of *r*, which is *r*!. This result is called combinations:

$latex _{n}{{C}_{r}}=\frac{{_{n}{{P}_{r}}n!}}{{r!}}$ |

By rewriting this formula, we can get the general combinations formula:

$latex _{n}{{C}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!r!}}$ |

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## Applying the permutations and combinations formulas

### EXAMPLE 1

Find the number of combinations if $latex n=10$ y $latex r=3$.

**Solution:** Simply, we can use the combinations formula replacing the values $latex n=10$ y $latex r=3$:

$latex _{n}{{C}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!r!}}$

$latex =\frac{{10!}}{{\left( {10-3} \right)!3!}}$

$latex =\frac{{10!}}{{\left( {7} \right)!3!}}$

$latex =\frac{{10\times 9\times 8\times 7!}}{{\left( 7 \right)!3!}}=120$

### EXAMPLE 2

Find the number of permutations if $latex n=10$ y $latex r=3$.

**Solution:** Again, we just have to use the permutations formula and replace the values $latex n=10$ y $latex r=3$:

$latex _{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}$

$latex =\frac{{10!}}{{\left( {10-3} \right)!}}$

$latex =\frac{{10!}}{{\left( {7} \right)!}}$

$latex =\frac{{10\times 9\times 8\times 7!}}{{\left( 7 \right)!}}=720$

### EXAMPLE 3

How many ways are there to choose a group of 5 people from a group of 12 people?

**Solution:** This is a problem of combinations, so we use the combinations formula with the values $latex n=12$ y $latex r=4$:

$latex _{n}{{C}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!r!}}$

$latex =\frac{{12!}}{{\left( {12-4} \right)!4!}}$

$latex =\frac{{12!}}{{\left( {8} \right)!4!}}$

$latex =\frac{{12\times 11\times 10\times 9\times 8!}}{{\left( 8 \right)!4!}}=495$

### EXAMPLE 4

How many ways are there to form a list of 4 desserts from a menu of 10 desserts?

**Solution:** Again, we just have to use the permutations formula and replace the values $latex n=10$ y $latex r=4$:

$latex _{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}$

$latex =\frac{{10!}}{{\left( {10-4} \right)!}}$

$latex =\frac{{10!}}{{\left( {6} \right)!}}$

$latex =\frac{{10\times 9\times 8\times 7\times 6!}}{{\left( 6 \right)!}}=5040$

### Try solving the following practice problems

## Frequently asked questions

### What is the main difference between combinations and permutations?

A permutation is an act of arranging items in order. Combinations are ways of selecting objects from a group in a way where the order of the objects does not matter.

### What are the formulas for combinations and permutations?

The permutations formula is $latex _{n}{{P}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!}}$.

The combinations formula is $latex _{n}{{C}_{r}}=\frac{{n!}}{{\left( {n-r} \right)!r!}}$.

### What is the relationship between permutations and combinations?

Permutations and combinations can be related by using the formula $latex _{n}{{C}_{r}}=\frac{{_{n}{{P}_{r}}n!}}{{r!}}$.

### What are examples of permutations and combinations?

Organizing digits, letters, people are examples of permutations.

Selecting objects from a menu, selecting people from a group are examples of combinations.

## See also

Interested in learning more about algebraic expressions, permutations, and combinations? Take a look at these pages:

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