Permutations and Combinations
Permutations and Combinations
Here \(n,r\) are integers with \(0\le r\le n\).
Permutations
A permutation of \(r\) distinct objects is an arrangement of the objects in a row in a specific order.
\[r!=r(r-1)(r-2)\cdots(2)(1),\qquad 0!=1\]
The number of permutations of \(r\) objects chosen from \(n\) distinct objects is
\[{}^nP_r=n(n-1)\cdots(n-r+1)=\frac{n!}{(n-r)!},\qquad 0\le r\le n\]
Combinations
A combination is a selection of \(r\) objects from \(n\) distinct objects when order does not matter.
\[{}^nC_r=\binom nr=\frac{n(n-1)\cdots(n-r+1)}{r!}=\frac{n!}{r!(n-r)!},\qquad 0\le r\le n\]
Useful relationships
\[{}^nP_r=\binom nr r!\]
\[\binom n0=\binom nn=1\]
\[\binom nr=\binom n{n-r}=\frac{n!}{r!(n-r)!}\]
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