The PERMUT function is used to return the total number of possible permutations for a given a set numbers of items where the order *matters *and *without *repetitions. In short, giving the possible number of ordered groupings for a given number of items, assuming every item can only be picked once.

## Syntax

`= PERMUT([`*number*],[*number_chosen*])

*number *= the number of items within a set

*number_chosen* = the number of items in each group

## Explanation

This function is part of the "Statistical" function group within Excel.

A permutation is a combination of numbers where the order matters. Think of this like playing pool and trying to sink all of the balls. Each ball can only be picked once, and the order in which they are sunk matters.

If you have a 15 balls, and you need to sink all 15 to win, you could use this formula to figure out the number of possible orders in which you could sink them all:

`= PERMUT(15,15) = 1,307,674,368,000`

This formula will return 1,307,674,368,000 which is the total possible number of ways to sink 15 balls, where the same number can't be picked multiple times.

What sets PERMUT apart from the PERMUTATIONA function is that PERMUT, *does not *count any permutations with repetition. If you are looking for repetition to be included, the PERMUTATIONA function is the one you'll want to use instead.

Here is a table to help you figure out which function you need to use:

The formula used to calculate this number is:

Where n = *number* and r = *number_chosen*

**Note:**

- The PERMUT function will return a #VALUE if any of the inputs are non-numerical.

- A #NUM error will be returned if either number is less than zero, as you can't have negative groupings

- A #NUM error will also be returned if the *number* < *number_chosen*,* *as the grouping must be smaller than the total set size.

- Any decimal number will be truncated

### Examples

## 1. How to Calculate Number of Possible Permutations Without Repetition

This is the standard use case for the PERMUT function. With a specific number of items and a specific group size, we can calculate all of the different ways in which the items in each set could be combined where order matters, without repetition.

Here the group size determines the number of permutations of pulling the first *n* objects. taking the 6,4 example below, there are 360 different ways that 4 objects could be pulled out of 6 total objects.

`= PERMUT(B3,C3)`

## 2. Calculate the Possible Number of Ways to Shuffle a Deck of Cards

Let's take the classic example of shuffling a deck of cards, and try to figure out how many possible ways you could shuffle it.

A standard deck of cards has 52 cards in it. Because each card can only exist in the deck once, we want to use the PERMUT function, which doesn't allow for repetition.

By using the PERMUT function, we can calculate that the number of possible permutations that a deck of cards could be shuffled, is 52!, or 52 * 51 * 50 * ... * 3 * 2 * 1. That's an 8 with 67 zeros behind it or:

80,658,175,170,943,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

`= PERMUT(52,52)`

## 3. Calculate the Number of Possible Ways a Horse Race Could be Won

For the third example, let's imagine a horse race is being run with 14 horses. If we want to know how many horses could get first, it would just be the original 14 horses. But how many ways are there for first and second place to come in from 14 horses?

This is important as in horse betting, there are a few different types of betting where guessing the first two, three, four, and five horses to cross the finish line is a popular type of bet. But what are the odds of each happening?

We're using 14 here as that is the commonly used limit for the number of horses running in each race, and assuming that the odds are equal across each horse ( which isn't actually true)

`= PERMUT(14,3)`