"Partial permutation" and "Combination with repetition"

What is "Partial permutation"

Partial permutation refers to selecting and arranging k elements from a set of n elements, where k is less than or equal to n. In a partial permutation, the order of selection matters, meaning that arranging the elements in different orders creates distinct outcomes. 

Example of Partial permutation

Let's say we have a basket of 5 different types of apples: {Red, Green, Yellow, Granny Smith, Fuji}. We want to select and arrange 3 apples from this basket in a specific order. (Please note that this "order" matters for Partial permutation.)

• 1: Red, 2: Green, 3: Yellow
• 1: Green, 2: Yellow, 3: Red
• 1: Yellow, 2: Red, 3: Green
• 1: Granny Smith, 2: Fuji, 3: Red
• 1: Fuji, 2: Granny Smith, 3: Green
• ...

etc. 

Actually there are 60 different ways to select and arrange 3 apples from the basket. This is the Partial permutation.

What is "Partial" in partial permutation

• (Not partial but full) permutation: You're arranging all the elements of a set.
• Partial permutation: You're selecting and arranging only a subset of the elements from a larger set.

How the partial permutation is calculated

Partial permutation, how many possible ways of selecting and arranging k elements from a set of n elements exist are often denoted by:

And it is calculated by:

.

This is equal to

.

So, when we select 3 apples out of 5 and arrange them in order, the number of possible partial permutations are:

What is "Combination with repetition"

"Combination with repetition" refers to selecting a certain number of objects from a set where the order of selection doesn't matter, and repetitions are allowed.

Example of combination with repetition

Let's say we have a basket of 5 different types of apples: {Red, Green, Yellow, Granny Smith, Fuji}. We want to select and arrange 3 apples from this basket. But the order doesn't matter this time, so, for example, {Green, Yellow, Fuji} and {Yellow, Green, Fuji} are considered to be same because the difference is only its order. 

• Red, Green, Yellow
• Granny Smith, Fuji, Red
• Yellow, Granny Smith, Green
• ...

etc. 

Actually there are 10 different ways to select 3 apples from the basket. This is the Combination with repetition.

How the combination with repetition is calculated

Combination with repetition (selecting k elements from a set of n elements) is often denoted by:

.

And it is calculated by:

.

And it is actually:

.

So, when we select 3 apples out of 5, the number of possible combinations are:

.

What if there is no disctintion among the apples

By the way... what if the 5 apples are all the same? In this case, whichever apple we choose, the result of 3 selections are always same, because there is no distinction among them.

{Fuji, Fuji, Fuji} out of {Fuji, Fuji, Fuji, Fuji, Fuji}

So there is only one way to select 3 apples in this case, which means, you don't need to even calculate to check how many possible combinations exist. 

References

Wikipedia, "Partial permutation", 2024 Apirl 30th visited