AI InterviewSearch…Search problems, topics, or pages…Ctrl KGet started

Topic

Permutations & Combinations aptitude questions

Counting arrangements and selections without listing them.

0 of 35 solved in this view

Formula sheetPermutations & Combinations: the essentialsFormula sheetPermutations & Combinations: the essentialsShow

  • Counting principle: independent stages multiply ('and'); separate cases add ('or').
  • n distinct objects arrange in n! ways. P(n, r) = n!/(n − r)! counts ordered selections and C(n, r) = n!/(r!(n − r)!) unordered ones, so P(n, r) = C(n, r) × r!.
  • C(n, r) = C(n, n − r), and C(n, 0) = C(n, n) = 1.
  • Repeated letters: n!/(p! × q! × …). BANANA gives 6!/(3! × 2!) = 60.
  • Circular: n people round a table sit in (n − 1)! ways; a necklace or garland, which can be turned over, in (n − 1)!/2.
  • Always together: glue the group into one unit, arrange, then multiply by the arrangements inside the unit. Never together = total − together.
  • Digits: fill the most restricted place first — a leading digit cannot be 0, and an even number needs an even units digit.
  • At least one = total − none. A set of n items has 2ⁿ subsets, 2ⁿ − 1 of them non-empty.
  • Handshakes, round-robin matches and lines through n points (no three in a line) are all C(n, 2) = n(n − 1)/2. An n-sided polygon has n(n − 3)/2 diagonals.
  • If order matters (signals, ranks, seats, numbers), use a permutation; if it does not (committees, teams), a combination.

Example: committees of 3 from 6 men and 3 women with at least one woman: C(9, 3) − C(6, 3) = 84 − 20 = 64.

Questions35

1

Arranging the letters of MANGO

How many different arrangements can be made from the letters of the word MANGO?

arrangements

30s

Easy2

Arranging the letters of BANANA

How many distinct arrangements are there of the letters of BANANA?

repeated letters

45s

Medium3

A committee with at least one woman

From 5 men and 4 women, how many committees of 3 can be formed that include at least one woman?

combinations · at least one

60s

Medium4

Three-digit numbers with distinct digits

How many three-digit numbers with all digits different can be formed using the digits 1 to 6?

digits

45s

Easy5

Six people around a round table

In how many ways can 6 people be seated around a round table?

circular

45s

Easy6

Handshakes at a party

At a party every person shakes hands exactly once with every other person, making 66 handshakes in all. How many people are there?

combinations

45s

Medium7

Arranging the letters of LEADER

In how many ways can the letters of the word LEADER be arranged?

arrangements

60s

Medium8

Arranging MATHEMATICS

How many distinct arrangements can be made of the letters of MATHEMATICS?

arrangements

90s

Hard9

Vowels always together

In how many ways can the letters of DETAIL be arranged so that the vowels occupy only odd positions?

arrangements · restrictions

105s

Hard10

Two letters always together

In how many ways can the letters of ORANGE be arranged so that the two vowels O and A are always together?

arrangements · restrictions

90s

Medium11

A committee of three

In how many ways can a committee of 3 people be chosen from 5 men and 4 women so that it has exactly 2 men?

combinations

60s

Medium12

At least two women

From 4 men and 3 women, in how many ways can a committee of 3 be formed with at least 2 women?

combinations

75s

Medium13

Counting the guests

At a party every guest shakes hands with every other exactly once, making 45 handshakes. How many guests were there?

combinations

60s

Medium14

Diagonals of an octagon

How many diagonals does a regular octagon have?

combinations · geometry

75s

Medium15

Three-digit numbers from 0 to 5

How many three-digit numbers with distinct digits can be formed using the digits 0, 1, 2, 3, 4, 5?

digits

90s

Hard16

Even numbers from given digits

How many three-digit even numbers with distinct digits can be formed from 1, 2, 3, 4, 5?

digits

75s

Medium17

Eight people at a round table

In how many ways can 8 people be seated around a circular table?

circular

45s

Easy18

Flowers in a garland

In how many ways can 6 different flowers be strung into a garland?

circular

75s

Medium19

Choosing from three colours

A bag has 6 red, 4 white and 5 blue balls. In how many ways can 3 balls of different colours be drawn?

combinations

60s

Medium20

Evaluating a combination

What is the value of C(10, 3)?

combinations

45s

Easy21

Evaluating a permutation

What is the value of P(6, 3)?

permutations

45s

Easy22

Books of two subjects

In how many ways can 4 maths books and 3 physics books be arranged on a shelf so that all the maths books stay together?

arrangements · restrictions

90s

Medium23

Selecting at least one item

In how many ways can one or more items be selected from 5 distinct items?

subsets

60s

Medium24

Arranging APPLE

In how many ways can the letters of APPLE be arranged?

arrangements

45s

Easy25

Arranging SUCCESS

How many distinct arrangements are there of the letters of SUCCESS?

arrangements

75s

Hard26

Choosing a team

In how many ways can a team of 4 be chosen from 9 players?

combinations

60s

Easy27

A team with a captain

From 8 players, in how many ways can a team of 3 be chosen and one of them named captain?

combinations

75s

Medium28

Four-digit even numbers

How many four-digit even numbers with distinct digits can be formed from 1, 2, 3, 4, 5?

digits

75s

Medium29

Keeping the vowels together

In how many ways can the letters of MOTHER be arranged so that the vowels are always together?

arrangements · restrictions

90s

Medium30

Selecting from two groups

A committee of 5 is to be chosen from 6 men and 4 women so that it contains exactly 3 women. In how many ways?

combinations

75s

Medium31

Handshakes among twelve

If 12 people each shake hands with every other exactly once, how many handshakes occur?

combinations

45s

Easy32

Seating with one fixed

In how many ways can 5 people be seated in a row if one particular person must sit at the left end?

arrangements · restrictions

45s

Easy33

A property of combinations

What is the value of C(12, 10)?

combinations

60s

Medium34

Signals from flags

How many different signals can be made using 3 flags out of 5 different flags, where order matters?

permutations

60s

Medium35

Counting subsets

How many subsets does a set of 6 elements have, including the empty set?

subsets

45s

Easy

Showing 35 of 35

The whole syllabus · Placement mock tests

AI InterviewGet started

Permutations & Combinations Aptitude Questions with Solutions

35 Permutations & Combinations questions with hints, worked solutions and time targets — 11 easy · 20 medium · 4 hard.

  • Section: Quantitative Aptitude
  • Questions: 35
  • By difficulty: 11 easy · 20 medium · 4 hard
  • Cost: Free, unlimited on every plan

Counting arrangements and selections without listing them.

Permutations & Combinations: the essentials

  • Counting principle: independent stages multiply ('and'); separate cases add ('or').
  • n distinct objects arrange in n! ways. P(n, r) = n!/(n − r)! counts ordered selections and C(n, r) = n!/(r!(n − r)!) unordered ones, so P(n, r) = C(n, r) × r!.
  • C(n, r) = C(n, n − r), and C(n, 0) = C(n, n) = 1.
  • Repeated letters: n!/(p! × q! × …). BANANA gives 6!/(3! × 2!) = 60.
  • Circular: n people round a table sit in (n − 1)! ways; a necklace or garland, which can be turned over, in (n − 1)!/2.
  • Always together: glue the group into one unit, arrange, then multiply by the arrangements inside the unit. Never together = total − together.
  • Digits: fill the most restricted place first — a leading digit cannot be 0, and an even number needs an even units digit.
  • At least one = total − none. A set of n items has 2ⁿ subsets, 2ⁿ − 1 of them non-empty.
  • Handshakes, round-robin matches and lines through n points (no three in a line) are all C(n, 2) = n(n − 1)/2. An n-sided polygon has n(n − 3)/2 diagonals.
  • If order matters (signals, ranks, seats, numbers), use a permutation; if it does not (committees, teams), a combination.

Example: committees of 3 from 6 men and 3 women with at least one woman: C(9, 3) − C(6, 3) = 84 − 20 = 64.

Questions

  1. Arranging the letters of MANGO Easy
    How many different arrangements can be made from the letters of the word MANGO?
  2. Arranging the letters of BANANA Medium
    How many distinct arrangements are there of the letters of BANANA?
  3. A committee with at least one woman Medium
    From 5 men and 4 women, how many committees of 3 can be formed that include at least one woman?
  4. Three-digit numbers with distinct digits Easy
    How many three-digit numbers with all digits different can be formed using the digits 1 to 6?
  5. Six people around a round table Easy
    In how many ways can 6 people be seated around a round table?
  6. Handshakes at a party Medium
    At a party every person shakes hands exactly once with every other person, making 66 handshakes in all. How many people are there?
  7. Arranging the letters of LEADER Medium
    In how many ways can the letters of the word LEADER be arranged?
  8. Arranging MATHEMATICS Hard
    How many distinct arrangements can be made of the letters of MATHEMATICS?
  9. Vowels always together Hard
    In how many ways can the letters of DETAIL be arranged so that the vowels occupy only odd positions?
  10. Two letters always together Medium
    In how many ways can the letters of ORANGE be arranged so that the two vowels O and A are always together?
  11. A committee of three Medium
    In how many ways can a committee of 3 people be chosen from 5 men and 4 women so that it has exactly 2 men?
  12. At least two women Medium
    From 4 men and 3 women, in how many ways can a committee of 3 be formed with at least 2 women?
  13. Counting the guests Medium
    At a party every guest shakes hands with every other exactly once, making 45 handshakes. How many guests were there?
  14. Diagonals of an octagon Medium
    How many diagonals does a regular octagon have?
  15. Three-digit numbers from 0 to 5 Hard
    How many three-digit numbers with distinct digits can be formed using the digits 0, 1, 2, 3, 4, 5?
  16. Even numbers from given digits Medium
    How many three-digit even numbers with distinct digits can be formed from 1, 2, 3, 4, 5?
  17. Eight people at a round table Easy
    In how many ways can 8 people be seated around a circular table?
  18. Flowers in a garland Medium
    In how many ways can 6 different flowers be strung into a garland?
  19. Choosing from three colours Medium
    A bag has 6 red, 4 white and 5 blue balls. In how many ways can 3 balls of different colours be drawn?
  20. Evaluating a combination Easy
    What is the value of C(10, 3)?
  21. Evaluating a permutation Easy
    What is the value of P(6, 3)?
  22. Books of two subjects Medium
    In how many ways can 4 maths books and 3 physics books be arranged on a shelf so that all the maths books stay together?
  23. Selecting at least one item Medium
    In how many ways can one or more items be selected from 5 distinct items?
  24. Arranging APPLE Easy
    In how many ways can the letters of APPLE be arranged?
  25. Arranging SUCCESS Hard
    How many distinct arrangements are there of the letters of SUCCESS?
  26. Choosing a team Easy
    In how many ways can a team of 4 be chosen from 9 players?
  27. A team with a captain Medium
    From 8 players, in how many ways can a team of 3 be chosen and one of them named captain?
  28. Four-digit even numbers Medium
    How many four-digit even numbers with distinct digits can be formed from 1, 2, 3, 4, 5?
  29. Keeping the vowels together Medium
    In how many ways can the letters of MOTHER be arranged so that the vowels are always together?
  30. Selecting from two groups Medium
    A committee of 5 is to be chosen from 6 men and 4 women so that it contains exactly 3 women. In how many ways?
  31. Handshakes among twelve Easy
    If 12 people each shake hands with every other exactly once, how many handshakes occur?
  32. Seating with one fixed Easy
    In how many ways can 5 people be seated in a row if one particular person must sit at the left end?
  33. A property of combinations Medium
    What is the value of C(12, 10)?
  34. Signals from flags Medium
    How many different signals can be made using 3 flags out of 5 different flags, where order matters?
  35. Counting subsets Easy
    How many subsets does a set of 6 elements have, including the empty set?

Placement tests that ask permutations & combinations