Mathematical tables: typical cobinatory related formulas
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.