Notes on everything: April 2024

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.