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This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !
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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. 
 Show source$$ 
Variations without repetition  yes ✓  no ✗  The same or less than as before operation. 
 Show source$$ 
Combinations without repetition  no ✗  no ✗  The same or less than as before operation. 
 Show source$$ 
Permutations with repetition  yes ✓  yes ✓  The same as before operation. 
 Show source$$ 
Variations with repetition  yes ✓  no ✗  The same, less or more as before operation. 
 Show source$$ 
Combinations with repetition  no ✗  no ✗  The same, less or more as before operation. 
 Show source$$ 
Factorial
Name  Formula  Legend 
Factorial  Show source$n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n1) \cdot n$   
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$n!^{(k)}=\begin{cases}n &, 0 < n \le k\\n \left((nk)!^{(k)}\right) &, n > k\end{cases}$   
Binomial and related formulas
Name  Formula  Legend 
Binomial coefficient  Show source$\binom{n}{k} = \frac{n!}{k!(nk)!}$   
Binomial expansion  Show source$\left(a + b\right)^n = \sum_{k=0}^{k=n} \binom{n}{k} \cdot a^{nk} \cdot b^k$   
Variations
Name  Formula  Legend 
Variations with repetition  Show source$\overline{V}_{n}^{k} = n ^ {k}$ 

Variations without repetition  Show source$V_{n}^{k} = \frac{n!}{(n  k)!}$ 

Combinations
Name  Formula  Legend 
Combinations with repetition  Show source$\overline{C}_{n}^{k} = \frac{(k + n  1)!}{k! (n  1)!}$ 

Combinations without repetition  Show source$C_{n}^{k} = \binom{n}{k} = \frac{n!}{k! (n  k)!}$ 

Permutations
Name  Formula  Legend 
Permutations with repetition  Show source$\overline{P}_{n}^{n1,n2,\dots,n_k} = \frac{n!}{n_1! \cdot n_2! \dots n_k!}$ 

Permutations without repetition  Show source$P_{n} = n!$ 

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 nelement set, the amount of its permutation is:
$P_{n} = n!$where:
 $P_{n}$  number of permutations without repetition of the nelement 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 occures more than once, then not all permutations are unique, e.g. swappning 1st and 3th letters in the word "eye" gives the same word. If we exclude nonunique 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 nelement 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 kth 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 ntuples.
 Variation without repetition is also called kpermutation 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 nelement 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},  2element 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},  2elements 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},  2element 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}.  2element 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}.
 permutations without repetition, simply we shuffle elements in all possible ways:
 ⓘ 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.
 combinatorial tables  short crib with common combinatorics related formulas,
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