Permutation and Combination — 24 Aptitude Questions and Answers with Explanations

Permutation and Combination — Questions and Answers

The single decision that determines every P&C answer is whether order matters. Arrangements (words, seatings, numbers) are permutations; selections (committees, teams, handshakes) are combinations. Repeated letters divide by the factorial of each repetition, and "always together" problems are solved by gluing the group into one unit and then arranging inside it.

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Permutation and Combination— Concepts, Formulas & Shortcuts

ⁿPᵣ = n!/(n − r)! for arrangements; ⁿCᵣ = n!/(r!(n − r)!) for selections.
Word with repeated letters: total arrangements = n! / (p! q! …) for each repeated letter.
"Always together": treat the group as one unit → (n − k + 1)! × k!.
"Never together" = total arrangements − arrangements with them together.
Circular arrangement of n objects = (n − 1)!; if clockwise and anticlockwise are the same, (n − 1)!/2.
Diagonals of an n-sided polygon = n(n − 3)/2.

Permutation and Combination Practice Questions with Answers

Attempt each question first, then open the explanation. All 24 questions below are free to read and require no signup.

Q1.In how many ways can the letters of the word "LEADER" be arranged?
Moderate
- A120
- B360
- C720
- D1,440
+Show Answer & Explanation
Answer: B. 360
Explanation: 6 letters with E repeated twice → 6!/2! = 720/2 = 360.
Permutation and Combination question 1 of 24
Q2.How many 3-letter words (no repetition) can be formed from 7 distinct letters?
Easy
- A35
- B120
- C210
- D343
+Show Answer & Explanation
Answer: C. 210
Explanation: ⁷P₃ = 7 × 6 × 5 = 210.
Permutation and Combination question 2 of 24
Q3.The value of ¹⁰C₃ is:
Easy
- A80
- B100
- C120
- D720
+Show Answer & Explanation
Answer: C. 120
Explanation: 10!/(3!·7!) = (10 × 9 × 8)/6 = 120.
Permutation and Combination question 3 of 24
Q4.In how many ways can 3 boys be chosen from 5 and 2 girls from 4?
Moderate
- A40
- B50
- C60
- D72
+Show Answer & Explanation
Answer: C. 60
Explanation: ⁵C₃ × ⁴C₂ = 10 × 6 = 60.
Permutation and Combination question 4 of 24
Q5.In how many ways can the letters of "OPTICAL" be arranged so that the vowels always come together?
Difficult
- A360
- B600
- C720
- D5,040
+Show Answer & Explanation
Answer: C. 720
Explanation: Vowels O, I, A form one unit → 5 units arranged in 5! = 120 ways, and the vowels among themselves in 3! = 6 → 720.
Permutation and Combination question 5 of 24
Q6.The number of diagonals in an octagon is:
Moderate
- A16
- B20
- C24
- D28
+Show Answer & Explanation
Answer: B. 20
Explanation: n(n − 3)/2 = 8 × 5/2 = 20.
Permutation and Combination question 6 of 24
Q7.In how many ways can 5 people be seated around a circular table?
Moderate
- A20
- B24
- C60
- D120
+Show Answer & Explanation
Answer: B. 24
Explanation: Circular arrangements = (5 − 1)! = 4! = 24.
Permutation and Combination question 7 of 24
Q8.How many 4-digit numbers can be formed using the digits 1–9 without repetition?
Moderate
- A2,016
- B3,024
- C4,536
- D6,561
+Show Answer & Explanation
Answer: B. 3,024
Explanation: 9 × 8 × 7 × 6 = 3,024.
Permutation and Combination question 8 of 24
Q9.The value of 5! is:
Easy
- A24
- B60
- C120
- D720
+Show Answer & Explanation
Answer: C. 120
Explanation: 5! = 5 × 4 × 3 × 2 × 1 = 120.
Permutation and Combination question 9 of 24
Q10.The value of ⁸C₂ is:
Easy
- A16
- B28
- C36
- D56
+Show Answer & Explanation
Answer: B. 28
Explanation: 8!/(2!·6!) = (8 × 7)/2 = 28.
Permutation and Combination question 10 of 24
Q11.In how many ways can the letters of the word "APPLE" be arranged?
Moderate
- A24
- B60
- C120
- D240
+Show Answer & Explanation
Answer: B. 60
Explanation: 5 letters with P repeated twice → 5!/2! = 120/2 = 60.
Permutation and Combination question 11 of 24
Q12.How many handshakes take place if 10 people each shake hands once with every other person?
Moderate
- A36
- B45
- C50
- D90
+Show Answer & Explanation
Answer: B. 45
Explanation: ¹⁰C₂ = (10 × 9)/2 = 45 handshakes.
Permutation and Combination question 12 of 24
Q13.How many 3-digit numbers can be formed from the digits 1 to 5 with repetition allowed?
Moderate
- A60
- B100
- C125
- D243
+Show Answer & Explanation
Answer: C. 125
Explanation: Each of the 3 places has 5 choices → 5³ = 125.
Permutation and Combination question 13 of 24
Q14.In how many ways can a committee of 3 be chosen from 8 people?
Easy
- A24
- B48
- C56
- D336
+Show Answer & Explanation
Answer: C. 56
Explanation: ⁸C₃ = (8 × 7 × 6)/6 = 56.
Permutation and Combination question 14 of 24
Q15.In how many ways can 6 people be seated in a row?
Easy
- A120
- B360
- C720
- D5,040
+Show Answer & Explanation
Answer: C. 720
Explanation: All 6 people are distinct and every ordering counts as different, so the total is 6! = 720.
Permutation and Combination question 15 of 24
Q16.How many words can be formed from the letters of "BANANA"?
Difficult
- A60
- B120
- C360
- D720
+Show Answer & Explanation
Answer: A. 60
Explanation: 6 letters with A three times and N twice → 6!/(3!·2!) = 720/12 = 60.
Permutation and Combination question 16 of 24
Q17.The value of 4! is:
Easy
- A12
- B16
- C24
- D48
+Show Answer & Explanation
Answer: C. 24
Explanation: 4 × 3 × 2 × 1 = 24.
Permutation and Combination question 17 of 24
Q18.The value of ⁶C₂ is:
Easy
- A12
- B15
- C20
- D30
+Show Answer & Explanation
Answer: B. 15
Explanation: (6 × 5)/2 = 15.
Permutation and Combination question 18 of 24
Q19.The value of ⁵P₂ is:
Easy
- A10
- B15
- C20
- D25
+Show Answer & Explanation
Answer: C. 20
Explanation: ⁿPᵣ = n!/(n − r)!, so ⁵P₂ = 5 × 4 = 20 ordered arrangements.
Permutation and Combination question 19 of 24
Q20.In how many ways can 3 prizes be given to 5 students, no student receiving more than one?
Moderate
- A15
- B30
- C60
- D125
+Show Answer & Explanation
Answer: C. 60
Explanation: ⁵P₃ = 5 × 4 × 3 = 60.
Permutation and Combination question 20 of 24
Q21.How many diagonals does a hexagon have?
Moderate
- A6
- B9
- C12
- D15
+Show Answer & Explanation
Answer: B. 9
Explanation: n(n − 3)/2 = 6 × 3/2 = 9.
Permutation and Combination question 21 of 24
Q22.In how many ways can the letters of "LEVEL" be arranged?
Difficult
- A20
- B30
- C60
- D120
+Show Answer & Explanation
Answer: B. 30
Explanation: 5 letters with L twice and E twice → 5!/(2!·2!) = 120/4 = 30.
Permutation and Combination question 22 of 24
Q23.In how many ways can 4 people be seated around a circular table?
Moderate
- A4
- B6
- C12
- D24
+Show Answer & Explanation
Answer: B. 6
Explanation: (4 − 1)! = 3! = 6.
Permutation and Combination question 23 of 24
Q24.The value of ⁿC₀ for any n is:
Easy
- A0
- B1
- Cn
- Dn!
+Show Answer & Explanation
Answer: B. 1
Explanation: There is exactly one way to choose nothing from n items.
Permutation and Combination question 24 of 24

Permutation and Combination — Frequently Asked Questions

How do I know whether to use permutation or combination?+

Ask whether swapping two chosen items creates a different outcome. Choosing a president and a secretary is a permutation; choosing a two-member committee is a combination.

Why divide by factorials for repeated letters?+

Because swapping two identical letters produces the same word. LEADER has 6 letters with E repeated twice, so 6!/2! = 360 distinct arrangements.

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