Permutations are thought of as positional arrangements, while combinations do not care about ordering, only the content of the set. It is useful to understand combinations and permutations in terms of "items" and "values". Items are unique entities that can only be used once, while values are freely reusable. This is often referred to as with or without "repitition".
Permutations
\[ n^r \]
Permutations of \( n \) values in \( r \) positions
\( A A, \: A B \)
\( B A, \: B B \)
\[ n! \]
Permutations of \( n \) items
\( A B C, \: C B A \)
\( A C B, \: B C A \)
\( B A C, \: C A B \)
\[ \frac{n!}{(n-r)!} \]
Permutations of \( n \) items in \( r \) positions
\( A B, \: B A \)
\( B C, \: C B \)
\( C A, \: A C \)
\[ \frac{n!}{\prod_{i = 1}^{n} m_{i}} \]
Permutations of \( n \) items, with \( m_i \) repitions allowed for each item i
\( m_A=2, \: m_B=1 \)
\( A A, \: A B\)
\( B A, \: \cancel{B B} \)
\[ \binom{n+r-1}{n} \]
Permutations of \( r \) integer values that sum to \( n \)
\( A+B=3, \: 0+3, \: 1+2, \: 2+1, \: 3+0 \)
Combinations
\[ \frac{n!}{r!(n-r)!} \]
Combinations of \( n \) items in \( r \) positions
\( A B, \: B C, \: C A \)
\[ \binom{n+r-1}{r} \]
Combinations of \( n \) values in \( r \) positions
\( A A, \: A B, \: B B, \: B C, \: C C, \: C A \)
\[ \binom{n-1}{r-1} \]
Combinations of \( r \) integer values that sum to \( n \)
\( A+B=3, \: 0+3, \: 1+2 \)
Shane Duffy
