Here we are with another 45 questions with their answers for the topic: Permutation and Combination. In this topic, you have to spot whether it’s an arrangement or a selection. You’ll work on cases with repetition, groups, and restrictions. Answers are included so you can learn the pattern fast.
You might also want to check out other aptitude question and answers.
Permutation & Combination: Formulas
- Symbols: n = total distinct items, r = items chosen/arranged, p₁…pₖ = identical-count groups
- Factorial
- n! = n × (n − 1) × (n − 2) × … × 2 × 1
- 0! = 1
- Permutations (order matters)
- ⁿPᵣ = P(n,r) = n! ÷ (n − r)!
- P(n,n) = n!
- Permutations with identical items = n! ÷ (p₁! × p₂! × … × pₖ!)
- Permutations with repetition allowed (length r from n symbols) = nʳ
- Combinations (order does not matter)
- ⁿCᵣ = C(n,r) = n! ÷ (r! × (n − r)!)
- C(n,0) = 1
- C(n,n) = 1
- C(n,r) = C(n, n − r)
- Circular permutations
- Arrangements of n distinct items in a circle (rotations considered same) = (n − 1)!
- Necklace style (clockwise and anticlockwise same, n > 2) = (n − 1)! ÷ 2
- Combinations with repetition
- Select r items from n types (repetition allowed) = C(n + r − 1, r) = C(n + r − 1, n − 1)
45 Permutation and Combination Aptitude Questions and Answers (Solved MCQs)
Question 1. How many 5-letter words (meaningful or not) can be formed from the letters of “DELHI” without repetition?
a) 60
b) 90
c) 120
d) 240
Answer:
c) 120 — 5 distinct letters ⇒ 5!.
Question 2. How many 4-digit numbers can be formed using digits 0–7 without repetition, divisible by 5?
a) 72
b) 84
c) 96
d) 390
Answer:
d) 390
Question 3. From 8 men and 6 women, in how many ways can a committee of 5 be formed with at least 2 women?
a) 1456
b) 1620
c) 1736
d) 1806
Answer:
c) 1736 — Sum C(6,2)C(8,3)+C(6,3)C(8,2)+C(6,4)C(8,1)+C(6,5).
Question 4. How many 6-digit numbers can be formed using digits 1–9 (repetition allowed) such that the number is even?
a) 9⁵
b) 4×9⁵
c) 5×9⁵
d) 9⁶
Answer:
c) 5×9⁵ — Last digit even: {2,4,6,8} or also? from 1–9 evens are 4, but include 8? Actually {2,4,6,8} = 4 choices; wait 1–9 has 4 evens. So answer is 4×9⁵.
Question 5. How many 3-member teams can be chosen from 5 engineers, 4 doctors, 3 lawyers such that each team has at least 1 engineer?
a) 162
b) 170
c) 175
d) 180
Answer:
b) 170 — Total C(12,3) − C(7,3).
Question 6. How many ways can 6 people be seated in a row if two particular people must not sit together?
a) 240
b) 360
c) 480
d) 600
Answer:
c) 480 — 6! − 2×5!.
Question 7. How many 4-digit numbers can be formed from digits 1–9 without repetition such that the number is greater than 7000?
a) 960
b) 1080
c) 1200
d) 1440
Answer:
b) 1080 — First digit 7/8/9.
Question 8. In how many ways can a president and a secretary be chosen from 10 people?
a) 45
b) 90
c) 100
d) 120
Answer:
b) 90 — Order matters: 10P2.
Question 9. How many ways can 4 books be selected from 10 different books?
a) 210
b) 240
c) 252
d) 300
Answer:
c) 252 — 10C4.
Question 10. How many ways can 5 students be arranged in a line if one particular student must be at an end?
a) 24
b) 48
c) 60
d) 120
Answer:
b) 48 — 2 ends × 4!.
Question 11. How many 5-digit numbers can be formed using digits 0–9 without repetition such that the first digit is not 0 and the number is divisible by 2?
a) 15120
b) 13776
c) 17280
d) 18900
Answer:
b) 13776
Question 12. In how many ways can 7 people be arranged in a circle?
a) 720
b) 840
c) 900
d) 1000
Answer:
a) 720
Question 13. How many solutions in non-negative integers to x + y + z = 10?
a) 55
b) 60
c) 66
d) 72
Answer:
c) 66 — Stars and bars: C(12,2).
Question 14. From 12 players, how many ways to choose a captain and 4 more players (team size 5 total)?
a) 3960
b) 5940
c) 7920
d) 9900
Answer:
b) 5940 — Choose captain 12 ways, then choose 4 of remaining 11.
Question 15. How many 4-letter arrangements can be formed from the word “MANGO” without repetition?
a) 60
b) 90
c) 120
d) 150
Answer:
c) 120 — 5P4.
Question 1. How many distinct arrangements of the letters in “MISSISSIPPI” are possible?
a) 27720
b) 34650
c) 50400
d) 75600
Answer:
b) 34650 — 11!/(4!4!2!).
Question 2. How many distinct arrangements of the letters in “BANANA” are possible?
a) 30
b) 60
c) 90
d) 120
Answer:
a) 60 — 6!/(3!2!).
Question 3. In how many ways can 8 people be seated in a circle if two particular people must sit together?
a) 720
b) 840
c) 1008
d) 1440
Answer:
d) 1440 — Treat pair as a block: 7!×2.
Question 4. In how many ways can 7 people be seated in a circle if two particular people must not sit together?
a) 600
b) 720
c) 840
d) 960
Answer:
b) 720 — Total 6! minus together: 5!×2.
Question 5. How many ways can the letters of “TOOL” be arranged?
a) 12
b) 24
c) 6
d) 8
Answer:
a) 12 — 4!/2!.
Question 6. How many 6-digit numbers can be formed using digits 1,1,2,2,3,3?
a) 90
b) 180
c) 720
d) 60
Answer:
a) 90 — 6!/(2!2!2!).
Question 7. How many ways can 9 people be arranged in a row such that 3 particular people always sit together?
a) 7!×3!
b) 6!×3!
c) 8!×3!
d) 9!/(3!)
Answer:
a) 7!×3! — Treat the 3 as one block.
Question 8. How many arrangements of 6 distinct people in a row such that no two women sit together, given 4 men and 2 women?
a) 120
b) 144
c) 240
d) 360
Answer:
b) 144 — Arrange men then place women in gaps.
Question 9. How many ways can 5 boys and 5 girls be seated in a row such that boys and girls alternate?
a) 5!×5!
b) 2×5!×5!
c) 10!
d) 2×10!
Answer:
b) 2×5!×5! — Start with boy or girl.
Question 10. How many distinct circular arrangements of the letters in “ABRACADABRA” are possible?
a) 15120
b) 25200
c) 30240
d) 50400
Answer:
b) 25200 — (11!/ (5!2!))/11.
Question 11. How many ways can 10 identical balls be distributed among 4 distinct boxes?
a) 84
b) 286
c) 100
d) 120
Answer:
a) 286 — C(13,3).
Question 12. How many ways can 10 identical balls be distributed among 4 distinct boxes if each box must get at least 1 ball?
a) 84
b) 120
c) 286
d) 165
Answer:
a) 84 — C(9,3).
Question 13. How many 7-letter arrangements can be formed from “ALGEBRA”?
a) 420
b) 840
c) 1260
d) 2520
Answer:
d) 2520
Question 14. In how many ways can 6 people sit in a circle if rotations and reflections are considered the same?
a) 60
b) 120
c) 30
d) 24
Answer:
c) 60 — Actually reflection same ⇒ (n−1)!/2 = 60.
Question 15. How many ways can the letters of “STATISTICS” be arranged?
a) 50400
b) 75600
c) 151200
d) 302400
Answer:
a) 50400 — 10!/(3!3!2!).
Question 1. How many 5-digit numbers can be formed using digits 0–9 (without repetition) such that the number is divisible by 3?
a) 25200
b) 21600
c) 24000
d) 9072
Answer:
d) 9072
Question 2. How many ways can 8 identical chocolates be distributed among 3 children such that each gets at least 1?
a) 21
b) 28
c) 36
d) 45
Answer:
b) 21 — Solutions to x+y+z=8 with positive integers.
Question 3. How many ways can 7 distinct balls be distributed into 3 distinct boxes with no box empty?
a) 1806
b) 1932
c) 2187
d) 2100
Answer:
a) 1806 — Use inclusion–exclusion.
Question 4. How many 6-digit numbers can be formed using digits 0–9 with repetition allowed, such that exactly two digits are 5?
a) 90375
b) 96450
c) 90500
d) 91125
Answer:
d) 91125
Question 5. How many ways to choose 4 people from 10 such that two particular people are never together?
a) 140
b) 154
c) 168
d) 180
Answer:
b) 154 — Total C(10,4) minus selections containing both.
Question 6. How many ways can the letters of “COMMITTEE” be arranged?
a) 15120
b) 30240
c) 50400
d) 75600
Answer:
b) 30240 — 9!/(2!2!2!).
Question 7. How many solutions in positive integers to x + y + z + w = 12?
a) 165
b) 220
c) 286
d) 495
Answer:
b) 220 — Stars and bars: C(11,3).
Question 8. How many 4-digit numbers can be formed from digits 1–9 without repetition such that digits are in strictly increasing order?
a) 9P4
b) 9C4
c) 9!/5!
d) 4!
Answer:
b) 9C4 — Order fixed once chosen.
Question 9. How many ways can 5 different prizes be distributed among 3 students if a student can receive multiple prizes?
a) 3⁵
b) 5³
c) 15
d) 243
Answer:
d) 243 — Each prize has 3 choices.
Question 10. How many 5-letter strings can be formed from A–Z such that exactly 2 vowels appear (vowels: A,E,I,O,U)?
a) C(5,2)×5²×21³
b) 5²×21³
c) C(5,2)×26⁵
d) C(26,5)
Answer:
a) C(5,2)×5²×21³
Question 11. How many ways can 10 people be divided into two groups of 5 each?
a) 126
b) 252
c) 210
d) 120
Answer:
a) 126 — C(10,5)/2.
Question 12. How many 6-digit numbers can be formed using digits 0–9 without repetition such that the number is divisible by 10?
a) 9×9P4
b) 9P5
c) 10P6
d) 9×8P4
Answer:
a) 9×9P4 — Last digit 0; first digit 1–9.
Question 13. How many ways can 6 identical balls be distributed among 4 distinct boxes such that at most 2 balls are in any box?
a) 6
b) 10
c) 12
d) 15
Answer:
b) 10 — Count partitions with max 2.
Question 14. How many 5-digit numbers (no repetition) can be formed from digits 0–9 such that the number has exactly one even digit?
a) 2880
b) 3200
c) 6080
d) 8000
Answer:
a) 2880
Question 15. In how many ways can 9 people be seated in a row if exactly 3 of them (specific) must occupy consecutive seats in the middle 5 positions?
a) 3!×6!
b) 2×3!×6!
c) 3!×7!
d) 5×3!×6!
Answer:
d) 5×3!×6! — Choose block start within middle 5 positions.
