PERMUTATIONA function

The PERMUTATIONA function in Excel calculates the number of permutations for a set of objects, where repetition of objects is allowed. Unlike the PERMUT function, which only considers arrangements without repetition, PERMUTATIONA allows for repeated selections of objects.

Syntax:

PERMUTATIONA(number, number_chosen)

Arguments:

• number: Required. The total number of items (objects) in the set.
• number_chosen: Required. The number of items to choose and arrange from the total set. With repetition allowed, the same item can be chosen more than once.

Output:

The function returns the number of permutations possible when choosing number_chosen items from a set of number items, allowing repetition.

Formula:

The formula for PERMUTATIONA is:

$$P(n, k) = n^k$$

Where:

• n is the total number of items in the set,
• k is the number of items chosen,
• The exponentiation (n^k) accounts for the repeated selections.

Example 1: Finding the Number of Permutations with Repetition

Suppose you have 5 objects and you want to find how many ways you can arrange 3 objects, allowing repetition.

Use the formula:

=PERMUTATIONA(5, 3)

This will return 125, meaning there are 125 possible ways to arrange 3 objects from a set of 5, allowing for repetitions.

Example 2: Finding Permutations with Repetition for a Larger Set

If you have 10 objects and you want to find how many ways you can arrange 4 objects, allowing repetition, use the formula:

=PERMUTATIONA(10, 4)

This will return 10,000, meaning there are 10,000 possible ways to arrange 4 objects from a set of 10, with repetition allowed.

Key Points:

• PERMUTATIONA calculates permutations where the order of the items matters and repetition is allowed.
• The result is higher than the standard permutation because repetitions are considered, and each item can appear multiple times in the arrangement.
• The number must be greater than or equal to number_chosen; otherwise, Excel will return an error.

Use Cases:

• Statistics: Calculate permutations when repetition is allowed, such as selecting items with replacement.
• Combinatorics: Solve problems where repeated selections are possible (e.g., rolling a die multiple times).
• Games: Determine the number of possible outcomes when items or elements can be repeated, such as drawing cards from a deck with replacement.

Notes:

• If number_chosen is greater than number, Excel will return an error (#NUM!).
• The PERMUTATIONA function is particularly useful in cases where the set allows for repetition (such as drawing from a bag of colored balls where each ball can be chosen more than once).