What is Permutations and Combinations Class 11: Definition & Basics

Understanding Permutations and Combinations in Class 11 Mathematics

In Class 11 NCERT Mathematics, Permutations and Combinations form a crucial chapter under the topic of Probability and Counting.

• Permutation refers to the arrangement of objects where the order is important.
• Combination refers to the selection of objects where the order does not matter.

This chapter introduces students to the concept of counting in a systematic way, enabling them to solve complex problems involving arrangements and selections efficiently. It is foundational for higher mathematics and competitive exams.

What is a Permutation? Definition and Formula

A permutation is an arrangement of objects in a specific order. When the order of objects matters, we use permutations.

• If you have $n$ distinct objects and want to arrange $r$ of them, the number of permutations is given by:

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

• Here, $n!$ (read as "n factorial") means the product of all positive integers up to $n$.

Example: How many ways can you arrange 3 books out of 5 on a shelf?

$$ _5P_3 = \frac{5!}{(5-3)!} = \frac{120}{2} = 60 $$

So, there are 60 different ways to arrange 3 books from 5.

What is a Combination? Definition and Formula

A combination is a selection of objects where the order does not matter.

• If you have $n$ distinct objects and want to select $r$ of them, the number of combinations is:

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

• This formula counts the number of ways to choose $r$ objects from $n$ without considering order.

Example: How many ways can you choose 3 students out of 5 for a team?

$$ _5C_3 = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10 $$

So, there are 10 ways to select the team.

Key Differences Between Permutations and Combinations

Understanding the difference between permutations and combinations is vital. Here's a comparison:

Remember: Permutations count ordered arrangements, combinations count unordered selections.

Factorials: The Building Block of Permutations and Combinations

The factorial function, denoted by $n!$, is essential in permutations and combinations.

• Definition:

$$ n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1 $$

• Special case:

$$ 0! = 1 $$

Factorials help calculate the total number of ways to arrange or select objects.

Example: Calculate $4!$:

$$ 4! = 4 \times 3 \times 2 \times 1 = 24 $$

Factorials simplify the permutation and combination formulas, making calculations straightforward.

Worked Example: Applying Permutations and Combinations

Example 1: How many 3-digit numbers can be formed using digits 1, 2, 3, 4 without repetition?

• Here, order matters (digits in different order form different numbers).
• Number of digits $n=4$, choose $r=3$.

$$ _4P_3 = \frac{4!}{(4-3)!} = \frac{24}{1} = 24 $$

So, 24 different 3-digit numbers can be formed.

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Example 2: From 5 students, how many ways can a group of 2 students be selected?

• Order does not matter.
• Number of students $n=5$, choose $r=2$.

$$ _5C_2 = \frac{5!}{2!3!} = \frac{120}{2 \times 6} = 10 $$

There are 10 ways to select the group.