Permutation and Combination
P and C (Permutation and Combination) is a fundamental branch of mathematics that deals with counting the number of ways in which objects can be arranged or selected. It is widely used in various fields such as probability, statistics, and computer science.
Permutation
- A permutation is an arrangement of objects in a specific order.
- The number of permutations of n objects taken r at a time is
given by: - P(n, r) = n! / (n-r)!
Combination
- A combination is a selection of objects without considering the order.
- The number of combinations of n objects taken r at a time is given by:
- C(n, r) = n! / (r! * (n-r)!)
Important Formulas
- nCr = nPr / r!
- nCr = nC(n-r)
- (n+r)Cr = nCr + nC(r-1)
Properties
- nC0 = nCn = 1
- nC1 = n
- nCr + nC(r-1) = (n+1)Cr
Applications
- Probability: Calculating the probability of events.
- Statistics: Analyzing data and making inferences.
- Computer science: Algorithm design and analysis.
Examples
- How many different ways can 5 people be seated in a row?
- P(5, 5) = 5! = 120
- How many different ways can a committee of 3 be formed from a group of 8 people?
- C(8, 3) = 8! / (3! * 5!) = 56
Additional Notes
- The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n.
- The
symbol ! is read as "factorial". - For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.