### Permutation Combination

**Permutation & Combination**

**Factorial of a number**

- The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. $$ n! = 1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$$

**Permutations of n distinct objects**

- The number of permutations of n distinct objects is n!.

**Permutations of n objects, r of which are alike**

- The number of permutations of n objects, r of which are alike, is given by: $$\frac{n!}{r!(n-r)!}$$

**Circular permutations**

- The number of circular permutations of n distinct objects is given by:(n-1)!

**Combinations of n distinct objects, r at a time**

- The number of combinations of n distinct objects, r at a time, is given by: $$ C(n,r) = \frac{n!}{r!(n-r)!} $$

**Combinations of n objects, r of which are alike**

- The number of combinations of n objects, r of which are alike, is given by: $$\frac{n!}{r_1! r_2! \cdots r_k!}$$ where $r_1, r_2, \cdots, r_k$ are the number of objects of each type and (r_1+r_2+\cdots+r_k=n)

**The relationship between permutations and combinations**

- The number of permutations of n objects, r of which are alike, is equal to the number of combinations of n objects, r at a time, multiplied by the number of permutations of r objects.

**Applications of permutations and combinations in probability and statistics**

- Permutations and combinations are used in probability and statistics to calculate the probability of events and to estimate population parameters.