# 1000 + Permutations and Combinations Aptitude Questions

This section will cover Multiple Choice Questions on Permutations and Combinations under Quantitative aptitude or Numerical ability. Target should be to complete all Questions with 80% accuracy. Most of the questions covered in this section has appeared in recent competitive exams. Regular practice of these sample questions should help you in achieving a good Test score.

#### Basic Concepts of Permutations and Combinations

We have covered few tips for solving Aptitude Questions on Permutations and Combinations. Students should go through them before attempting Test series.

• Permutations formula: Permutations is defined as arrangement of r things that can be done out of total n things. This is denoted by nPwhich is equal to n!/(n-r)!
• Combinations formula
1. Combinations is defined as selection of r things that can be done out of total n things. This is denoted by nCwhich is equal to n!/r!(n-r)!
2. As per the Fundamental Principle of Counting, if a particular thing can be done in ways and another thing can be done in ways, then either one of the two can be done in m + n ways and both of them can be done in m × n ways.

Solved Permutations and Combinations Problems

Example 1: How many four-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6 (Repetition of digits not allowed)?

Solution: Thousand’s place can be filled in 6 ways. Hundred’s place can be filled in 5 ways. Ten’s place can be filled in 4 ways. Unit’s place can be filled in 3 ways. So, using the Fundamental Principle of Counting, we get the answer as 6 × 5 × 4 × 3 = 360. Or using the formula of Permutations, we need to arrange 4 digits out of total 6 digits. This can be done in 6P4= 360 ways.

Example 2: A person has 6 friends to be invited for dinner through invitation cards, and he has 3 servants. In how many ways can he extend the invitation card?

Solution: We can see that the 1st friend has 3 options to receive the card, i.e. either from 1st servant or 2nd or 3rd. Similarly, 2nd friend also has 3 options to receive the card, i.e. either from 1st servant or 2nd or 3rd. So we can say that each of the 6 friends has 3 options to receive the card. Hence the answer would be 3 × 3 × 3 × 3 × 3 × 3 = 36=729 ways.

Example 3: How many words can be formed from the letters of the word TRIANGLE with T always at the beginning and E at the end.

Solution: If we fix up T in the beginning and E at the end, then the remaining 6 letters can be arranged in 6P6 that is 6! which is 720 ways.

Example 4: How many words can be formed with the letters of the word “ORDINATE” so that vowels occupy odd places ?

Solution : There are 4 vowels and 4 consonants in the word “ORDINATE”. We have to arrange 8 letters in such a way that vowels occupy odd places i.e. 1,3,5 and 7. So four vowels can be arranged in these 4 odd places as 4P4 or 4! . And remaining 4 consonants can occupy 4 positions left in 4P4 way.

So, total ways or permutations will be 4! X 4! = 576

Example 5: In how many ways 5 boys and 3 girls can be seated in a row that no 2 girls are together ?

Solution: 5 boys can be seated in 5P5 ways. So now 3 girls can seat in between boys so that they are not together.

__ B __ B __ B __ B __ B__

So now 3 girls can occupy 6 places which is 6P3 ways.

Hence total ways are 5P5 * 6P3 = 14400

Example 6: How many 3 digit numbers between 400 and 1000 can be formed with the digits 0, 2, 3, 4, 5, 6 if no digit is repeated ?

Solution: Since the number is greater than 400, so hundred’s place can be filled by 4 , 5 or 6 only (3 ways) . Now unit and ten’s place can be filled by remaining 5 digits i.e. 5P2.

so total ways will be 5 * 5P2 = 60

Example 7: How many different words can be formed with the letters of the word MISSISSIPPI ?

Solution: Total 11 characters are there in the word. Out of which 4 S, 4 I, 2 P are repeated. So in such problems we will be dividing total ways by repeating ways i.e 11!/(4!)(4!)(2!) = 34650

Example 8: In how many ways a committee of 5 members can be selected from 6 men and 5 women, consisting of 3 men and 2 women ?

Solution 8: 3 men out of 6 can be selected in 6C3 ways. 2 women out of 5 ways can be selected in 5C2 ways. So 6C3 * 5C2 = 200 ways.

Example 9: A team of 3 members need to be formed out of 5 men and 2 women. In how many ways team can be formed so as to include at least 1 woman ?

Solution 9: Team of 3 can be formed in 2 ways

a) 2 men and 1 woman – 5C2 * 2C1

b) 1 man and 2 women – 5C1 * 2C2

So total ways will be (5C2 * 2C1) + (5C1 * 2C2) = 25 ways.

#### Permutations and Combinations MCQ –

Q 1 – There are 12 Buses running between London and Manchester, In how many ways can Jose Mourinho go from London to Manchester and return in a different Bus ?
• 132
• 144
• 264
• 64

Q 2 – In how many ways number of 2 digits can be formed out of the four digits 1,2,3 and 4 ?”
• 6
• 9
• 12
• 18

Q 3 – Find the number of words that can be formed using letters L,M,N and O ?”
• 32
• 64
• 72
• 128

Q 5 – In how many ways a football Team of 11 can be selected from a squad of 15 players ?
• 2100
• 2330
• 2410
• 2730

Q 6 – How many arrangements can be formed out of letters of the word CALCUTTA ?
• 2520
• 10080
• 5040
• 6000

Q 7 – How many different words can be formed with the letters of the word BHARAT?
• 180
• 240
• 300
• 360

Q 8 – How many numbers greater than a million can be formed with the digits 2,3,0,4,3,3,3 ?
• 300
• 360
• 440
• 620

Q 9 – In how many ways the letters of the word FAMILY can be arranged when F and Y are always together ?
• 60
• 120
• 240
• 480

Q 10 – In how many different ways the seven letters in the word MINIMUM be arranged if all of the 7 letters are used each time ?
• 36
• 128
• 420
• 626

Q 11 – If a Car registration number have four symbols, the first two of which are letters and the remainder digits, how many different registration numbers can be made ?
• 58500
• 62250
• 63375
• 65515

Q 12 – How many 5 digit cell phone numbers with pairwise distinct digits can be composed ?
• 28560
• 30240
• 16650
• 21424

Q 13 – How many greeting cards can be purchased by chain of 12 friends on the eve of Diwali festival if each sends a card to each other ?
• 66
• 96
• 132
• 166

Q 14 – A gentleman has got 6 sorts of note papers, 7 different ink-stands and 4 different pens. In how many ways can he begin to write a letter ?
• 168
• 176
• 186
• 196

Q 15 – There are 5 ways from A to B and 3 from B to C, How many ways are there from A to C via B ?
• 5
• 10
• 12
• 15

Q 16 – In a crossword puzzle there are 2 solutions to each of the 3 given places and 3 solutions to 1 other place. How many different solutions can be set in ?
• 12
• 24
• 36
• 48

Q 17 – How many words can be formed from the letters of the word GLOBE ?
• 60
• 120
• 150
• 180

Q 18 – Find the number of Triangles formed by joining the angular points of a polygon of 9 sides ?
• 27
• 45
• 60
• 84

Q 19 – In how many different ways can the letters of the word Auction can be arranged in such a way that the vowels always come together ?”,
• 30
• 48
• 144
• 576

Q 20 – In an examination there are 3 multiple choice questions and each question has 4 choices. The number of ways in which a student can fail to get all answers correct is ?
• 11
• 27
• 63
• 84

For more Numerical ability or Quantitative Aptitude Questions, please refer here.