Tables show common formulas useful in combinatorics such as number of variations (with or without repetition) or binomial.

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Summary#

Type of operation	Elements order matters	Keeps all elements	Number of items after operation	Algorithm how to create	Formula
Permutations without repetition	yes ✓	yes ✓	The same as before operation.	We shuffle items.	Show source $-$
Variations without repetition	yes ✓	no ✗	The same or less than as before operation.	We choose selected items, and we build new sequence from them.	Show source $-$
Combinations without repetition	no ✗	no ✗	The same or less than as before operation.	We choose selected items.	Show source $-$
Permutations with repetition	yes ✓	yes ✓	The same as before operation.	We shuffle items, and we ignore non-unique results.	Show source $-$
Variations with repetition	yes ✓	no ✗	The same, less or more as before operation.	We choose selected items, we clone some of them (if we want), and we build new sequence.	Show source $-$
Combinations with repetition	no ✗	no ✗	The same, less or more as before operation.	We choose selected items, and we clone some of them (if we want).	Show source $-$

Factorial#

Name	Formula	Legend
Factorial	Show source $n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n-1) \cdot n$	-
Factorial (recursive definition)	Show source $n!=\begin{cases}1 &, n < 2\\n \cdot (n - 1)! &, n \ge 2\end{cases}$	-
Doubel factorial (recursive definition)	Show source $n!!=\begin{cases}1 &, n < 2\\n \cdot (n - 2)!! &, n \ge 2\end{cases}$	-
Multifactorial (recursive definition)	Show source $n!^{(k)}=\begin{cases}n &, 0 < n \le k\\n \left((n-k)!^{(k)}\right) &, n > k\end{cases}$	-

Binomial and related formulas#

Name	Formula	Legend
Binomial coefficient	Show source $\binom{n}{k} = \frac{n!}{k!(n-k)!}$	-
Binomial expansion	Show source $\left(a + b\right)^n = \sum_{k=0}^{k=n} \binom{n}{k} \cdot a^{n-k} \cdot b^k$	-

Variations#

Name	Formula	Legend
Variations with repetition	Show source $\overline{V}_{n}^{k} = n ^ {k}$	$\overline{V}_{n}^{k}$ - number of variations with repetition (it may be for example number of 3-letter words built upon 26 possible letters i.e. aaa, aab, aba etc.), $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words), $k$ - number of items used (it may be for example length of the word or number of balls pulled out from the bucket).
Variations without repetition	Show source $V_{n}^{k} = \frac{n!}{(n - k)!}$	$V_{n}^{k}$ - number of variations without repetition (it may be for example number of 3-letter words built upon 26 possible letters, but each letter can be used only once i.e. abc, abd, dac etc.), $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words), $k$ - number of items used (it may be for example length of the word or number of balls pulled out from the bucket).

Combinations#

Name	Formula	Legend
Combinations with repetition	Show source $\overline{C}_{n}^{k} = \frac{(k + n - 1)!}{k! (n - 1)!}$	$\overline{C}_{n}^{k}$ - number of combinations with repetition, $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words), $k$ - number of items used (it may be for example length of the word or number of balls pulled out from the bucket).
Combinations without repetition	Show source $C_{n}^{k} = \binom{n}{k} = \frac{n!}{k! (n - k)!}$	$C_{n}^{k}$ - number of combinations without repetition, $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words), $k$ - number of items used (it may be for example length of the word or number of balls pulled out from the bucket).

Permutations#

Name	Formula	Legend
Permutations with repetition	Show source $\overline{P}_{n}^{n1,n2,\dots,n_k} = \frac{n!}{n_1! \cdot n_2! \dots n_k!}$	$\overline{P}_{n}^{n1,n2,\dots,n_k}$ - number of permutations with repetition of the n-element sequence, $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words), $n_1$ - frequency of appearance of the first element, $n_2$ - frequency of appearance of the second element, $n_k$ - frequency of appearance of the k-th element (e.g. the frequency of the letter "e" in the word "eye" is 2).
Permutations without repetition	Show source $P_{n} = n!$	$P_{n}$ - number of permutations without repetition of the n-element sequence, $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words).

Some facts#

Permutation consists in changing the order of elements in the sequence. Colloquially, we can say that permutation is a mixing of elements.
The permutation result includes the same number of elements as the source set.
If we have a n-element set, the amount of its permutation is:

$P_{n} = n!$
where:
- $P_{n}$ - number of permutations without repetition of the n-element sequence,
- $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words).
If some elements in original set occurs more than once, then not all permutations are unique, e.g. swappning 1-st and 3-th letters in the word "eye" gives the same word. If we exclude non-unique words, then the amount of permutation is:

$P_{n} = n!$
where:
- $\overline{P}_{n}^{n1,n2,\dots,n_k}$ - number of permutations with repetition of the n-element sequence,
- $n$ - number of items in the pool (it may be for example number of alphabet letters, which we use to create words),
- $n_1$ - frequency of appearance of the first element,
- $n_2$ - frequency of appearance of the second element,
- $n_k$ - frequency of appearance of the k-th element (e.g. the frequency of the letter "e" in the word "eye" is 2).
Variation consists in choosing any number of elements from the pool and then building a new sequence from them.
Sequence length after variation can be different than a source sequence. Depending on whether the element can be used again or not, it can be longer, shorter or the same as the length of the original one.
If the number of elements in the pool is n, and we choose k elements, the number of possible variations is:

$V_{n}^{k} = \frac{n!}{(n - k)!}$
or if we assume that the same element (e.g. the letter of the alphabet) we can use more than once:

$\overline{V}_{n}^{k} = n ^ {k}$
Variation with repetition is also called n-tuples.
Variation without repetition is also called k-permutation of n.
Combination consists in choosing any number of elements from the pool but without building a new sequence. We simple pull out selected items from the pool and... its all.
In the case of the combination the order of the elements does not matter. It is only important if the given element is in use or not (e.g. whether a given number was drawn in the lottery).
If we have the n-element set and we choose k elements, then the number of possible combinations is:

$C_{n}^{k} = \binom{n}{k} = \frac{n!}{k! (n - k)!}$
or if we assume that the same element can be used more than once:

$\overline{C}_{n}^{k} = \frac{(k + n - 1)!}{k! (n - 1)!}$
ⓘ Example: Suppose we have a set of numbers: $\{1,2,3,4\}$ . Examples permutations, combinations or variations of this set are:
- permutations without repetition, simply we shuffle elements in all possible ways:
  {1,2,3,4},
  {2,1,3,4},
  {3,1,2,4},
  {1,3,2,4},
  {2,3,1,4},
  {3,2,1,4},
  {3,2,4,1},
  {2,3,4,1},
  {4,3,2,1},
  {3,4,2,1},
  {2,4,3,1},
  {4,2,3,1},
  {4,1,3,2},
  {1,4,3,2},
  {3,4,1,2},
  {4,3,1,2},
  {1,3,4,2},
  {3,1,4,2},
  {2,1,4,3},
  {1,2,4,3},
  {4,2,1,3},
  {2,4,1,3},
  {1,4,2,3},
  {4,1,2,3},
- 2-element variations without repetition, we choose 2 elements and arrange them into a new sequence, the elements order matters:
  {1,2}, {1,3}, {1,4},
  {2,1}, {2,3}, {2,4},
  {3,1}, {3,2}, {3,4},
- 2-elements variations with repetitions, as above but we can use the same number more than once:
  {1,1}, {1,2}, {1,3}, {1,4},
  {2,1}, {2,2}, {2,3}, {2,4},
  {3,1}, {3,2}, {3,3}, {3,4},
  {4,1}, {4,2}, {4,3}, {4,4},
- 2-element combinations without repetition, we just pull out 2 numbers from the pool and we do not do... nothing. We don't arrange them into another sequence, so the elements order does not matter:
  {1,2}, {1,3}, {1,4},
  {2,3}, {2,4}
  {3,4}.
- 2-element combinations with repetitions, as above, but the same number can be used more than once:
  {1,1}, {1,2}, {1,3}, {1,4},
  {2,2}, {2,3}, {2,4}
  {3,3}, {3,4},
  {4,4}.

ⓘ Hint: More combinatorial items on Calculla:
- combinatorial tables - short crib with common combinatorics related formulas,
- permutations generator - simple tool to create list of all possible permutations (with or without repetition) based on given input pool of items,
- combinations generator - simple tool to create list of all possible combinations (with or without repetition) based on given input pool of items,
- variations generator - simple tool to create list of all possible variations (with or without repetition) based on given input pool of items.

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