Permutation and Combination
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Table of Content:
Permutation and Combination Formulas
Definitions
Permutation: Arrangement of objects in a specific order.
Combination: Selection of objects without considering the order.
Factorial
\[
n! = n \times (n-1) \times (n-2) \times \cdots \times 1
\]
Permutation Formulas
Permutations of \( n \) Objects
\[ P(n) = n! \]Permutations of \( n \) Objects Taken \( r \) at a Time
\[ P(n, r) = \frac{n!}{(n-r)!} \]Combination Formulas
Combinations of \( n \) Objects Taken \( r \) at a Time
\[ C(n, r) = \frac{n!}{r! (n-r)!} \]Properties of Permutations and Combinations
Symmetry in Combinations
\[ C(n, r) = C(n, n-r) \]Repetition in Permutations
\[ P(n, r) \text{ with repetition} = n^r \]Special Cases
Permutation of Like Objects
If there are \( n \) objects with \( p_1 \) of one kind, \( p_2 \) of another kind, ..., \( p_k \) of \( k \) kinds, the permutations are:
\[ P(n; p_1, p_2, \ldots, p_k) = \frac{n!}{p_1! \cdot p_2! \cdots p_k!} \]Circular Permutations
The number of ways to arrange \( n \) objects in a circle:
\[ P_{\text{circular}}(n) = (n-1)! \]Example Problems
Example 1: Finding Permutations
If there are 5 books and 3 are to be arranged on a shelf, how many arrangements are possible?
\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{120}{2} = 60 \]Example 2: Finding Combinations
If there are 6 fruits and 2 are to be selected, how many selections are possible?
\[ C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{720}{2 \cdot 24} = 15 \]