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 Combination**s

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 bywhich is equal to n!/(n-r)!^{n}P_{r }**Combinations formula**- Combinations is defined as selection of r things that can be done out of total n things. This is denoted by
^{n}C_{r }which is equal to n!/r!(n-r)! - As per the Fundamental Principle of Counting, if a particular thing can be done in
*m*ways and another thing can be done in*n*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.

- Combinations is defined as selection of r things that can be done out of total n things. This is denoted by

**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 ^{6}P_{4}= 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 1^{st }friend has 3 options to receive the card, i.e. either from 1^{st }servant or 2^{nd }or 3^{rd}. Similarly, 2nd friend also has 3 options to receive the card, i.e. either from 1^{st }servant or 2^{nd }or 3^{rd}. 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 = 3^{6}=**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 ^{6}P_{6} 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 ^{4}P_{4} or 4! . And remaining 4 consonants can occupy 4 positions left in ^{4}P_{4} 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 ^{5}P_{5} 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 ^{6}P_{3} ways.

Hence total ways are ^{5}P_{5} * ^{6}P_{3} = 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. ^{5}P_{2}.

so total ways will be 5 * ^{5}P_{2} = 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 ^{6}C_{3} ways. 2 women out of 5 ways can be selected in ^{5}C_{2} ways. So ^{6}C_{3} * ^{5}C_{2} = 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 – ^{5}C_{2} * ^{2}C_{1}

b) 1 man and 2 women – ^{5}C_{1} *^{ 2}C_{2}

So total ways will be (^{5}C_{2} * ^{2}C_{1}) + (^{5}C_{1} *^{ 2}C_{2}) = 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

Answer is “A”

**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

Answer is “C”

**Q 3 – Find the number of words that can be formed using letters L,M,N and O ?”**

- 32
- 64
- 72
- 128

Answer is “B”

**Q 5 – In how many ways a football Team of 11 can be selected from a squad of 15 players ?**

- 2100
- 2330
- 2410
- 2730

Answer is “D”

**Q 6 – How many arrangements can be formed out of letters of the word CALCUTTA ?**

- 2520
- 10080
- 5040
- 6000

Answer is “C”

**Q 7 – How many different words can be formed with the letters of the word BHARAT?**

- 180
- 240
- 300
- 360

Answer is “D”

**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

Answer is “B”

**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

Answer is “C”

**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

Answer is “C”

**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

Answer is “A”

**Q 12 – How many 5 digit cell phone numbers with pairwise distinct digits can be composed ?**

- 28560
- 30240
- 16650
- 21424

Answer is “B”

**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

Answer is “C”

**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

Answer is “A”

**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

Answer is “D”

**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

Answer is “B”

**Q 17 – How many words can be formed from the letters of the word GLOBE ?**

- 60
- 120
- 150
- 180

Answer is “B”

**Q 18 – Find the number of Triangles formed by joining the angular points of a polygon of 9 sides ?**

- 27
- 45
- 60
- 84

Answer is “D”

**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

Answer is “D”

**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

Answer is “C”

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