Permutations and Combinations – Class 11 Maths
The chapter Permutations and Combinations is a crucial part of Class 11 Mathematics. It introduces methods to count arrangements and selections, which is fundamental in probability, statistics, and combinatorial problems.
📖 Factorials
The factorial of a non-negative integer n is:
n! = n × (n − 1) × (n − 2) × … × 1
Special cases:
- 0! = 1
- 1! = 1
Factorials are used to calculate permutations and combinations.
🔹 Permutations
Permutation refers to the arrangement of objects in a specific order.
1. Permutation of n distinct objects:
n!
2. Permutations of r objects taken from n:
nPr = n! / (n − r)!
3. Permutations when some objects are identical:
n! / (p! q! r! …)
Where p, q, r are repetitions of identical objects.
Example:
How many ways can the letters of the word “LEVEL” be arranged?
Here, L repeats 2 times, E repeats 2 times:
Number of arrangements = 5! / (2! × 2!) = 30
🔹 Combinations
Combination refers to selection of objects where order does not matter.
1. Combination of r objects from n:
nCr = n! / [r! × (n − r)!]
Example:
From 5 students, select 2 to represent the class:
Number of ways = 5C2 = 10
🔹 Relation Between Permutations and Combinations
nPr = nCr × r!
This formula connects the two concepts and is frequently used in problems.
🔹 Important Points and Rules
- Factorials are used extensively in counting.
- Permutation = Arrangement (order matters)
- Combination = Selection (order does not matter)
- Repetition rules must be carefully applied when objects are identical.
- Always check whether order matters before solving a problem.