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 Thinking Operations Binomial theorem⦿ Binomial theorem Combinatorics⦿ Permutations⦿ Combination⦿ Variation Sets⦿ Operations on Sets Logic⦿ Logical Operations and Truth Tables

Combinatorics

# Permutations

## Permutation without Repetition

A permutation is an arrangement, or listing, of n distinct objects in which the order is important.

Total number of permutations in case of n elements:

${P}_{n}=n·\left(n-1\right)·\left(n-2\right)·...·2·1=n!$

Example:

In case of 4 elemts: {a,b,c,d}: $n=4,{P}_{4}=4!=4·3·2·1=24$

## Permutation with Repetition

A permutation is an arrangement, or listing, of n objects in which the order is important. The elements are repeated. Number of repetations:

${k}_{1},{k}_{2},{k}_{3},...,{k}_{r};\left({k}_{1}+{k}_{2}+{k}_{3}+...+{k}_{r}\le n\right)$

Total number of permutations:

${P}_{n}^{{k}_{1},{k}_{2},{k}_{3},...,{k}_{r}}=\frac{n!}{{k}_{1}!·{k}_{2}!·{k}_{3}!·...·{k}_{r}!}$

Example:

In case of 7 elemet: {a,a,a,a,b,b,c} first element repeats 4 times, second element repeats 2 times: $n=7,{k}_{1}=4,{k}_{2}=2,{k}_{1}=1$

Total number of permutations:

${P}_{7}^{4,2,1}=\frac{7!}{4!·2!·1!}=105$