SOLUTION: what is the difference between nCk and nPk

Question 1113913: what is the difference between nCk and nPk
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52848)   (Show Source): You can put this solution on YOUR website!
.

nCk is the number of Combinations of n items taken k at a time. = . nPk is the number of Permutations of k items taken from the set of n elements = n*(n-1)*(n-2)* . . . * (n-k+1). We consider combinations when we select the groups of k items from the set of n items without looking on their order (when the order does not matter). We consider permutations when we select the groups of k items from the set of n items and consider these groups as ordered sets (when the order does matter). Again, or one more time: nCk is the number of all subsets of k elements of a given set of n elements. nPk is the number of all ordered subsets of k elements of a given set of n elements.

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On Combinations and Permutations see the lessons
    - Introduction to Permutations
    - PROOF of the formula on the number of Permutations
    - Problems on Permutations
    - Introduction to Combinations
    - PROOF of the formula on the number of Combinations
    - Problems on Combinations

    - OVERVIEW of lessons on Permutations and Combinations
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
nCk is the combination formula.
nPk is the permutation formula.

combination formula does not take order into consideration.
permutation formula does.

combination formula is nCk = n! / (k! * (n-k)!)

permutation formula is nPk - n! / (n-k)!.

that k! in the denominator of the nCk formula is what removes consideration of order within each set.

here's a simple example:

consider the set {abcd}

n = 4
k = 2

this means you want to get all possible sets of 2 elements each from a set of 4 elements.

4C2 = 4! / (2! * 2!) = 6

4P2 = 4! / 2! = 12

4C2 sets would be:

ab
ac
ad
bc
bd
cd

4P2 sets would be:

ab
ac
ad
bc
bd
cd

ba
ca
da
cb
db
dc

the 4C2 results consider ab and ba as members of the same set, since order doesn't matter within each set.

the 4P2 results consider ab as one set and ba as another separate set, since order within the set does matter.

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