Permutation and Combination – Notes for Competitive Exams

Introduction

Permutation and Combination is an important topic in Quantitative Aptitude for exams like SSC, Banking, Railway, Defence, and State Exams.

It helps solve problems related to arrangements, selections, and counting. Questions are generally easy to moderate if formulas are memorized.

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Key Concepts

1. Permutation (Arrangements)

• Permutation refers to the arrangement of objects in a specific order.

Formula:

• Permutation of n objects taken r at a time:

$$^nP_r = \frac{n!}{(n-r)!}$$nPr​=(n−r)!n!​

Where $n! = n \times (n-1) \times … \times 1$n!=n×(n−1)×…×1

• Permutation of n objects all arranged:

$$^nP_n = n!$$nPn​=n!

• Permutation with repeated objects:

$$\frac{n!}{p_1! \, p_2! \, p_3! \, …}$$p1​!p2​!p3​!…n!​

Where $p_1, p_2, …$p1​,p2​,… are repetitions of identical items.

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2. Combination (Selections)

• Combination refers to selecting objects where order does not matter.

Formula:

• Combination of n objects taken r at a time:

$$^nC_r = \frac{n!}{r!(n-r)!}$$nCr​=r!(n−r)!n!​

• Relation between permutation and combination:

$$^nP_r = ^nC_r \times r!$$nPr​=nCr​×r!

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Common Types of Questions

1. Linear Arrangements: Arranging objects in a line.
2. Circular Arrangements: Arranging objects in a circle.
3. Selections: Choosing a group from a larger set.
4. Repeated Objects: Arrangements or selections with identical items.
5. Special Conditions: Arrangements with restrictions (e.g., two people together or apart).

Permutation and Combination Questions

Q1. How many ways can 5 people sit in a row?

Q2. How many ways can 5 people sit in a circle?

Q3. How many ways can 3 students be chosen from 10?

Q4. How many 3-digit numbers can be formed using the digits 1, 2, 3, 4, 5 without repetition?

Q5. How many ways can the letters of the word “LEVEL” be arranged?

Q6. How many 4-digit numbers divisible by 5 can be formed using digits 1, 2, 3, 4, 5?

Q7. In how many ways can 7 people be seated in a row if 2 particular people must sit together?

Q8. How many ways can 7 people be arranged in a row if 2 particular people cannot sit together?

Q9. How many ways can 4 men and 3 women be arranged in a row if all women sit together?

Q10. How many ways can a committee of 3 men and 2 women be chosen from 5 men and 4 women?

Q11. How many 3-digit numbers can be formed using digits 0, 1, 2, 3, 4 if repetition is allowed?

Q12. How many words can be formed from the letters of the word “BANANA”?

Q13. How many ways can 6 people be arranged in a circle?

Q14. How many ways can 4 books be arranged on a shelf?

Q15. How many ways can a team of 5 be selected from 10 players?

Q16. How many ways can 3 letters be chosen from the word “SUCCESS”?

Q17. How many 3-digit even numbers can be formed using digits 1, 2, 3, 4, 5?

Q18. How many ways can 4 people be seated in a row if 2 must not sit together?

Q19. How many ways can 8 people be seated around a circular table?

Q20. How many ways can 2 boys and 2 girls be chosen from 5 boys and 4 girls?

Q21. How many ways can 4 letters be selected from the word “PROBABILITY”?

Q22. How many arrangements are possible with the letters of the word “MISSISSIPPI”?

Q23. How many ways can 3 prizes be awarded to 10 students if no student receives more than one prize?

Q24. How many ways can 5 different books be given to 3 students if each student receives at least one book?

Q25. How many ways can 6 people be arranged in a row if 3 of them must sit together?

Answer

Answers

Q1. 5! = 120
Q2. (5–1)! = 4! = 24
Q3. 10C3 = 120
Q4. 5 × 4 × 3 = 60
Q5. 5! / (2! × 2!) = 30
Q6. 4 × 5 × 5 × 2 = 200
Q7. 6! × 2 = 720 × 2 = 1440
Q8. 7! – (6! × 2) = 5040 – 1440 = 3600
Q9. 4! × 3! = 24 × 6 = 144
Q10. 5C3 × 4C2 = 10 × 6 = 60
Q11. 4 × 5 × 5 = 100 (since first digit ≠ 0)
Q12. 6! / (3! × 2! ×1!) = 60
Q13. (6–1)! = 5! = 120
Q14. 4! = 24
Q15. 10C5 = 252
Q16. 7C3 = 35
Q17. 3 × 4 × 5 = 60
Q18. 4! – 3! = 24 – 6 = 18
Q19. (8–1)! = 7! = 5040
Q20. 5C2 × 4C2 = 10 × 6 = 60
Q21. 11C4 = 330
Q22. 11! / (4! × 4! × 2! ×1!) = 34650
Q23. 10P3 = 720
Q24. 5C3 × 3! + 5C4 × arrangements … = 150 (can expand if needed)
Q25. 4! × 3! = 24 × 6 = 144