Permutation and Combination: Complete Guide with Formulas and Examples

Permutation and combination are fundamental concepts in mathematics, especially in the field of probability and statistics. These two tools are widely used in solving problems related to arrangements, selections, probability, and algebra. While both deal with the concept of counting, the key difference lies in whether the order of items matters.

What is Permutation?

A permutation is an arrangement of items in a particular order. If the order of arrangement matters, then we are dealing with permutations.

Permutation Formula:

• For arranging n different items:

  $$P(n)=n!$$

• For selecting and arranging r items out of n:

  P(n,r)=n!(n−r)!P(n, r) = \frac{n!}{(n - r)!}P(n,r)=(n−r)!n!​

Example of Permutation:

How many ways can 3 books be arranged on a shelf out of 5?

P(5,3)=5!(5−3)!=1202=60 waysP(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60 \text{ ways}P(5,3)=(5−3)!5!​=2120​=60 ways

What is Combination?

A combination is a selection of items without considering the order. If the order doesn’t matter, then it's a combination.

Combination Formula:

C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n - r)!}C(n,r)=r!(n−r)!n!​

Example of Combination:

How many ways can we choose 3 books from a set of 5?

C(5,3)=5!3!⋅(5−3)!=1206⋅2=10 waysC(5, 3) = \frac{5!}{3! \cdot (5-3)!} = \frac{120}{6 \cdot 2} = 10 \text{ ways}C(5,3)=3!⋅(5−3)!5!​=6⋅2120​=10 ways

Permutation vs Combination: Tabular Comparison

Real-Life Applications

Permutation Applications:

• Ranking participants in a competition

• Arranging digits for passwords

• Seating arrangements

Combination Applications:

• Forming a committee

• Choosing lottery numbers

• Picking questions in an exam

Common Mistakes to Avoid

1. Confusing order relevance: Always check if the problem asks for arrangements (permutation) or selections (combination).

2. Misapplying factorial rules: Remember that $0!=1$.

3. Using permutation instead of combination in probability: For cases like drawing cards, combinations are usually more appropriate.

Advanced Concepts

Repetition in Permutations:

When some elements are repeated:

P=n!p1!⋅p2!⋅…⋅pk!P = \frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!}P=p1​!⋅p2​!⋅…⋅pk​!n!​

Combination with Repetition:

Formula:

$$C(n+r-1,r)$$

Example: Choosing 3 candies from 5 types, allowing repeats:

$$C(5+3-1,3)=C(7,3)=35$$

Tips for Competitive Exams

• Use shortcut formulas and tricks.

• Understand the context to decide between permutation and combination.

• Memorize factorials up to at least 10.

Frequently Asked Questions (FAQs)

Q1: How do I know when to use permutation or combination?
A: Use permutations when order matters, like rankings. Use combinations when order doesn’t matter, like choosing a team.

Q2: What does nCr mean?
A: It stands for the number of combinations of r items from a total of n, calculated as:

C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n - r)!}C(n,r)=r!(n−r)!n!​

Q3: Can permutation be greater than combination?
A: Yes. Since permutation considers different orders, it always results in a greater or equal value compared to combination.

Q4: Is 0! equal to 1? Why?
A: Yes. By definition and for consistency in formulas, $0!=1$.

Q5: Are there real-life scenarios where both are used together?
A: Yes, for example in card games or multi-round tournaments where you first choose a team (combination) and then arrange them (permutation).

Conclusion

Permutation and combination are powerful mathematical tools that play a critical role in solving complex counting problems. While they may seem similar at first glance, understanding the difference in whether order matters is crucial for applying them correctly. With formulas, practical examples, and real-life relevance, mastering these concepts can significantly boost your problem-solving skills, especially in competitive exams.