Syllabus Unit 4 – Combinatorics and Probability Combinatorics Difference Between Permutation and Combination

Difference Between Permutation and Combination

Master the difference between Permutation and Combination for CBSE Class 11 Applied Mathematics. Learn formulas, key properties, and solved examples.

Fundamental Concept of Permutation

A permutation is an arrangement of a set of objects in a specific order. When the order of selection matters, such as arranging books on a shelf or forming a password, we use permutations. The number of ways to arrange r objects from a set of n distinct objects is denoted by nPr.

nPr = n! / (n − r)!
Example: Solved Example: Arranging 3 letters from {A, B, C, D}
Show Step-by-Step Solution

Step 1: Identify n=4 and r=3.
Step 2: Apply formula 4P3 = 4! / (4 − 3)!.
Step 3: Calculate 24 / 1 = 24.
Answer: 24 ways.

Fundamental Concept of Combination

A combination is a selection of objects where the order does not matter. When we choose a committee or a team from a larger group, the arrangement within the group is irrelevant, so we use combinations. The number of ways to select r objects from n distinct objects is denoted by nCr.

nCr = n! / (r!(n − r)!)
Example: Solved Example: Selecting 2 players from 5
Show Step-by-Step Solution

Step 1: Identify n=5 and r=2.
Step 2: Apply formula 5C2 = 5! / (2!(5 − 2)!).
Step 3: Calculate 120 / (2 × 6) = 10.
Answer: 10 ways.

The Relationship Formula

The primary difference is that permutations count order, while combinations do not. Because every combination of r objects can be arranged in r! ways, the number of permutations is always r! times the number of combinations. This leads to the fundamental identity connecting the two.

nPr = nCr × r!
Example: Solved Example: Verify 5P2 = 5C2 × 2!
Show Step-by-Step Solution

Step 1: Calculate 5P2 = 20.
Step 2: Calculate 5C2 = 10.
Step 3: Calculate 10 × 2! = 10 × 2 = 20.
Answer: 20 = 20, verified.