A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. Permutation and combination is all about counting and arrangements made from a certain group of data.

1. Evaluate __50!__

47!

a. 102500

b. 112584

c. 117600

d. 118450

2. Find the value of ^{85}P_{3 }.

a. 565350

b. 595650

c. 535950

d. 565350

3. Find the value of (^{20}C_{18})*(^{20}C_{20})

a. 400

b. 380

c. 360

d. 350

4. How many words with or without meaning, can be formed by using all the letters of the word, ‘ORANGE’, using each letter exactly once?

a. 700

b. 720

c. 750

d. 800

5. There are 28 stations between Ernakulam and Chennai. How many second class tickets have to be printed, so that a passenger can travel from one station to any other station?

a. 800

b. 820

c. 850

d. 870

6. In how many ways can the letters of the word, ‘TECHNOLOGY’ be arranged?

a. 1804400

b. 1814400

c. 1714400

d. 1704400

7. A bag contains 2 yellow balls, 3 white balls and 5 red balls. In how many ways can two balls be drawn from the bag?

a. ^{2}C_{2}

b.^{10}C_{2}

c.^{8}C_{2}

d.^{5}C_{2}

8. In how many ways can the letters of the word, ‘LANGUAGE’ be arranged in such a way that the vowels always come together?

a. 600

b. 700

c. 720

d. 750

9. In how many ways can the letters of the word, ‘KEYBOARD’ be arranged in such a way that the vowels always come together?

a. 4250

b. 4520

c. 4320

d. 4230

10. In how many ways can a team 16 be chosen out of a batch of 20 players?

a. 4845

b. 6852

c. 3125

d. 5846

11. How many ways can the letters of the word, ‘MACHINE’ be arranged so that the vowels may occupy only the odd positions?

a. 210

b. 576

c.144

d. 456

12. From a group of 5men and 4 women, 3 persons are to be selected to form a committee so that at least 2 men are there are on the committee. In how many ways can it be done?

a. 20

b. 50

c. 65

d. 86

13. In how many ways can a committee consisting of 4 men and 5 women be formed from a group of 7 men and 9 women?

a. ^{7}C_{4 } ^{9}C_{5}

b. ^{4}C_{7 } ^{5}C_{9}

c. ^{7}C5 ^{9}C_{4}

d. ^{9}C_{4 } ^{7}C_{5}

14. In how many ways can 5 boys and 3 girls sit around a table in such a way that no two girls sit together?

a. 1000

b. 1400

c.1440

d. 1800

Directions for questions 15 to 16: Refer the data below and answer the questions below:

A letter lock has 3 rings each containing 6 letters.

15. What is the maximum number of false trials that can be made before the lock is opened?

a. 3*^{26}C_{6}

b. (^{26}C_{6})^{3}

c. ^{26}C_{6}. 3!

d. 215

16. How many such three letter passwords can exist?

a. 216

b. ^{26}C_{6} *3

c. (^{26}C_{6})^{3}

d. (^{26}C_{6})^{3 }*6^{3}

17. How many different words can be formed from the word DAUGHTER so that ending and beginning letters are consonants?

a. 7200

b. 14400

c. 360

d. 1440

18. Out of 6 consonants and 3 vowels, how many words of 4 consonants and 2 vowels can be formed?

a. 1050

b. 25200

c. 32400

d. 5800

19. A box contains 3 white balls, 4 black balls and 5 yellow balls. In how many ways can 4 balls be drawn from the box, if at least one yellow ball is to be included in the draw?

a. 652

b. 547

c.425

d. 356

20. In how many ways can 22 books on English and 20 books on Hindi be placed in a row on a shelf so that two books on Hindi may not be together?

a. 4586

b. 5896

c. 2415

d. 1771

**Answer & Explanations**

1. Evaluate __50!__ = __50*49*48* (47!)__ = 50*49*48 = 117600

47! 47!

2.^{ 85}P_{3 }= __85!__ = __85!__ = __85*84*83*82!__ = 85*84*83 = 595650

(85-3)! 82! 82!

3.^{ 20}C_{20 }= 1

(^{20}C_{2})*(^{20}C_{20}) = __20! __* 1= __20*19*18!__ = 20*19*1= 380

18! 18!

4. Exp: The word ‘ORANGE’ contains 6 different letters.

Therefore, Required number of words= Number of arrangement of 6 letters, taken all at a time

=^{6}P_{6} = 6!= 6*5*4*3*2*1= 720

5. Exp: The total number of stations= 30

From 30 Stations we have to choose any two stations and the direction of travel (Ernakulam to Chennai

is different from Chennai to Ernakulam) in ^{30}P_{2} ways.

^{ 30}P_{2}= 30*29= 870

6. Exp: The word ‘TECHNOLOGY’ contains 10 letters namely 2O, 1T, 1E, 1C, 1H, 1N, 1L, 1G, 1Y.

Therefore, Required number of ways=__ 10! __= __ 10! __ __ __ (2!)(1!) (1!) (1!) (1!) (1!) (1!) (1!) (1!) 2!

= __10*9* 8*7*6*5*4*3*2*1__ = 1814400

2*1

7. Exp: Total number of balls= 2+3+5 = 10

2 balls can be drawn from 10 balls in ^{10}C_{2}

8. Exp: In the word ‘LANGUAGE’ we treat the vowels AUAE as one letter. Thus, we have LNGG (AUAE).

This we have 5 letters of which G occurs 2 times and the rest are different.

Number of ways arranging these letters= __5! __=5*4*3= 60

2!

Now, 4 letters of which A occurs 2 times and the rest are different, can be arranged in __4!__ = 4*3=12.

2!

Therefore, Required number of ways= 60*12= 720

9. Exp: In the word ‘KEYBOARD’ we treat the vowels EOA as one letter. Thus, we have KYBRD (EOA).

Thus we have 6 letters can be arranged in 6! = 720 ways

The vowels (EOA) can be arranged among themselves in 3! = 6 ways

Therefore, Required number of ways= (720*6)= 4320

10. Exp: Required number of ways= ^{20}C_{16}= ^{20}C(_{20}-_{16)} = ^{20}C_{4}

= __20*19*18*17__ = 4845.

4*3*2*1

11. Exp: In the word ‘MACHINE’ 3 vowels and 4 consonants.

v v v v

__ _ __ _ __ _ __

Now, 3 vowels can be placed at any of 3 places, out of which 4 marked 1,3,5,7.

Number of ways arranging the vowels= ^{4}P_{3} = (4*3*2) =24

Also, 4 consonants at the remaining 4 positions may be arranged in = ^{4}P_{4} = 4!= 24 ways.

Therefore, Required number of ways= (24*24) =576.

12. We have (2men and 1 woman) or (3men only)

Therefore, Required number of ways= (^{5}C_{2}*^{4}C_{1}) + (^{5}C_{3})

= __5*4__ * 4 + ^{5}C_{2}

2*1

= 10*4 + 10

= 40 +10 =50

13. Exp: Group consisting of 7 men and 9 women

4 men can be selected from 7 men in ^{7}C_{4 }ways

5 women can be selected from 9 women in ^{9}C_{5 }ways

Therefore, Total number of ways= ^{7}C_{4 } ^{9}C_{5}

14. Exp: The 5 boys can be seated around a table in 4! Ways. In between them there are 5 places.

The 3 girls can be placed in the 5 places in ^{5}P_{3} ways.

Therefore, Required number of ways= 4!* ^{5}P_{3}

=24*60=1440

15. Exp:

Maximum possible permutation of letters= 6*6*6= 216

Out of 216 different permutations only 1 is correct.

Maximum number of false trials= 216-1= 215

16. Exp: 1^{st} ring: 6 out of 26 alphabets can be selected in ^{26}C_{6} ways.

And is for 2^{nd} and 3^{rd} ring.

Also, these 3 set of 6 letters can be arranged amongst themselves in 6^{3} ways.

Hence, total number of 3 letter passwords = ^{26}C_{6 }* ^{26}C_{6 }*^{26}C_{6} *6^{3 }ways.

17. Exp: Here total letters are 8,3 vowels and 5 consonants. Here 2 consonants can be chosen in ^{5}C_{2} ways

and these 2 consonants can be put it in 2! Ways. The remaining 6 letters can be arranged in 6! Ways.

The words beginning and ending letters with consonant = ^{5}C_{2} *2! *6! = 14400

18. Number of ways of selecting (4 consonants out of 6) and (2 vowels out of 3)

= ^{6}C_{4} * ^{3}C_{2}

= ^{6}C_{2} * ^{3}C_{1}

= __6*5__ *3

2*1

= 15*3= 45

Number of groups, each having 4 consonants and 2 vowels= 45.

Each group consists of 6 letters.

Number of ways of arranging 6 letters among themselves

= 6! = (6*5*4*3*2*1)= 720

Therefore, Required number of words= 45*720= 32400

19. Exp: We may have (1 yellow and 3 others) or (2 yellow and 2 others) or (3 yellow and 1 others) or (4

yellow).

Therefore, Required number of ways= (^{4}C_{1}*^{8}C_{3})+ (^{4}C_{2}*^{8}C_{2})+ (^{4}C_{3}*^{8}C_{1})+ (^{4}C_{4})

= 4* __8*7*6 __ + __4*3__ * __8*7__ + (^{4}C_{1}*8) +1

3*2*1 2*1 2*1

= 224+168+32+1= 425.

20. Exp: In order that two books on Hindi are never together, we must place all these books as under:

H E H E H E H….. H E H

Where H denotes the position of Hindi book and E that of English book.

Since there are 22 books on English, the number of places marked E are 23.

Now, 20 places out of 23 can be chosen in ^{23}C_{20} = ^{23}C_{3 }= __23*22*21__

3*2*1

= 1771 ways.

Hence the number of ways = 1771 ways

A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. Permutation and combination is all about counting and arrangements made from a certain group of data.
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What is a formula of permutation?
n!/(n-r)!
What is the difference between permutation and combination?
In permutation order of things is considered, whereas in combination, order of things doesnt matter.
Example of permutation: Picking first, second and third place winners. Example of combination: Picking three winners.