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!} $ $$\frac{n!}{r!(n-r)!}$$

• 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!(n-r)!} $ $$\frac{n!}{r_1! r_2! \cdots r_k!}$$ 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.