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# MELJUN CORTES Permutation

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MELJUN CORTES Permutation

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### MELJUN CORTES Permutation

1. 1. PERMUTATIONS AND COMBINATIONSMELJUN CORTESMeljun Cortes 1 5 DECEMBER 2008
2. 2. My words..... My purpose here is to give a few questions on fundamentals of permutations and combinations. Iwelcome your suggestions. I also request you to help me in spreading social entrepreneurship across theglobe – for which I need support of you people – notof any VIP. With your help, I can spread the ideas – for which we stand.... Meljun Cortes 2 5 DECEMBER 2008
3. 3. What is permutation & combination? Permutation = it denotes order / Sequence but combination = it only denotes that some objects are togetherexample : ABC can have only one combination taking all of them together. But permutations are many : - ABC,ACB,BCA,BAC,CBA,CAB Meljun Cortes 3 5 DECEMBER 2008
4. 4. What is formula of permutation ? Npr = n! / (n-r)! p=permutation n= total number of objects r=how many objects you are taking at a time! = multiply with reducing numbers till it reaches 1 example : 5p5 = 5! / (5-5)! 5!=5*4*3*2*1 0! = 1 thus answer = 120 answerMeljun Cortes 4 5 DECEMBER 2008
5. 5. There are 5 books on maths, 3 on entrepreneurship and 2 on economics. In how many ways can we arrange them. We have to keep books of a subject together. There are 3 subjects so we can arrange them in 3! methods = 6there are 5 books on maths, we have 5! methods = 120there are 3 books on entrepreneurship, so we have 3! methods there are 2 books on economics, so we have 2! methods. Total we have : 6*120*6*2 = 8640 answer Meljun Cortes 5 5 DECEMBER 2008
6. 6. A court has given 6 to 3 decision infavaour of an issues. In how many ways can it reverse the decision? There are 9 judges, 5 will make a majority there may be 5,6,7,8,9 judges against the issuewe have 9c5 or 9c6 or 9c7 or 9c8 or 9c9 methods : 126 + 84+36+9+1 =256 answerMeljun Cortes 6 5 DECEMBER 2008
7. 7. Three persons go into a room which has 8 seats, in how many ways can they occupy the seats ?First person has 8 options, 2nd person has 7 options and 3rd person has 6 options. Total = 8*7*6 = 336 answer Meljun Cortes 7 5 DECEMBER 2008
8. 8. There are 5 routes between Bikaner and Delhi.You may choose any route. In how many ways can you go and return from Delhi? 5*5 = 25Meljun Cortes 8 5 DECEMBER 2008
9. 9. How many permutations are possible from ACCOUNTANT ?There are 10 digits, there are 2A, 2C, 2T, 2N, so we have : 10! / (2!*2!*2!*2!) answerMeljun Cortes 9 5 DECEMBER 2008
10. 10. How many different 4 digit letters can you make out of A,B,C,D,E? N = 5 (A,B,C,D,E) R=4 formula = Npr = n! / (n-r)! =5!/(5-4)! = 120 answerMeljun Cortes 10 5 DECEMBER 2008
11. 11. How many different 4 digit numbers can you make out of 1,2,3,4,0? N = 5 (1,2,3,4,0) R=4 but 0 cannot come in the first digit for first digit we have 4 options (1,2,3,4), for next digits, we can use 0. thus we have 4*4*3*2*1 = 96 options OR formula = Npr = n! / (n-r)! =5!/(5-4)! but this contains all those numbers whichstart with 0. so let us keep 0 as fixed for 1st digit and solve it. Now we have to pick up 3 digit out of 4 contd.Meljun Cortes 11 5 DECEMBER 2008
12. 12. contd..... If it is not 0, permutation will be : formula = Npr = n! / (n-r)! =5!/(5-4)! = 120 Zero fixed for 1st potion, we have these options : Npr = n! / (n-r)! n=4,r=3 4!/(4-3)! = 24 deduct this 24 from 120 120 -24 = 96 answer you can use any formula (out of these 2), you get the sameMeljun Cortes answer 12 5 DECEMBER 2008
13. 13. How many different 4 digit numbers canyou make out of 1,2,3,4,0 which are divisible by 2? Start with 96 of the last question now pick up all those which are ending with 1 : 3*3*2*1 = 18 similarly those which are ending with 3 3*3*2*1 = 18 thus 96 – (18+18) = 60 seems to be the answerMeljun Cortes 13 5 DECEMBER 2008
14. 14. In how many ways can Raj invite any 3 of his 7 friends? This is a question of combination. Here order(sequence) is not important, his friends can come in any order. Thus this is a case of combination. Formula : N! / ((n-r)!*r!) you can calculate combination by dividing permutation by r! =7! / ((7-3)!*3!) =(7*6*5)/(3*2*1) = 35 answerMeljun Cortes 14 5 DECEMBER 2008
15. 15. How many different words can you frame from FUTURE ? Here we have two U total we have 6 digits. Formula : N ! / L! N= total number of digits L = those digits which are repeated. Answer = 6! / 2! = 360 answerMeljun Cortes 15 5 DECEMBER 2008
16. 16. How many different words can you frame from DALDA ? Here we have two D & A total we have 5 digits. Formula : N ! / L! N= total number of digits L = those digits which are repeated. Answer = 5! / (2!*2!) = 30 answerMeljun Cortes 16 5 DECEMBER 2008
17. 17. In how many ways can 8 person sit around a round table ?For questions relating to round table , we have to use the following formula : (n-1)! So here answer = (8-1)! = 7! =5040 answerMeljun Cortes 17 5 DECEMBER 2008
18. 18. How many 4 digit numbers canbe formed out of 1,2,3,5,7,8,9 if no digit is repeated. Total number ofdigits = 7 formula = Npr n =7 r 4 7p4 = 7! / 3! =7*6*5*4 = 840Meljun Cortes 18 5 DECEMBER 2008
19. 19. How many numbers greaterthan 2000 can be formed from 1,2,3,4,5. No repeatition is allowed. 5 digit numbers = 5! = 1204 digit numbers,: we cant take 1 in the beginning. We have 4 options for 1st digit 4 for 2nd digit 3 for 3rd digit ... 4*4*3*2*1 = 96 total = 216 answerMeljun Cortes 19 5 DECEMBER 2008
20. 20. There are 6 books on english, 3 on maths, 2 on GK. In how many ways can they be placed in shelf, if books of 1 subject are together? We have 3 subjects so 3! books of same subjects can be interchanged. So answer : 3!*6!*3!*2! =6*720*6*2 = 51840 answerMeljun Cortes 20 5 DECEMBER 2008
21. 21. How many words can we make out of DRAUGHT, the vowels are never separated? Number of vowels = 2 other digits = 5 we will treat vowels as 1 word so we have 6!. Vowels can be interchanged so 2! so answer = 6!*2! = 1440 answerMeljun Cortes 21 5 DECEMBER 2008
22. 22. In how many ways can 8 pearls be used to form a necklace ?In questions of necklace, we use the following formula : ½ (N-1)! Here we can take reverse order of left to right or right to left, so divide by ½ =1/2 (8-1)! =2520Meljun Cortes 22 5 DECEMBER 2008
23. 23. In how many number of ways can 7 boys form a ring ? (7-1) ! = 6! = 720 answerMeljun Cortes 23 5 DECEMBER 2008
24. 24. 50 different jewels can be set to form necklace in how many ways ? ½ ( n -1) ! = ½ (50 -1)! =1/2 (49)!Meljun Cortes 24 5 DECEMBER 2008
25. 25. How many number of different digits can be formed from0,2,3,4,8,9 between 10 to 1000? Let us assume that repeatition is not allowed Let us make 2 digit numbers : for first digit we have 5 option, for 2nd digit also we have 5 options (including 0) = 25 for 3 digit numbers : 5*5*4 = 100 total 125 if repeatition is allowed : for 2 digit : 5 * 6 = 30 for 3 digit : 5*6*6 = 180 total = 210 answerMeljun Cortes 25 5 DECEMBER 2008
26. 26. What is the number of permutations of 10 different things taking 4 at a time in which one thing never comes ? =9p4 = (9*8*7*6) =3024Meljun Cortes 26 5 DECEMBER 2008
27. 27. There are 5 speakers (A,B,C,D,E) , in how many ways can we arrange their speach that A always speaks before BFor A and then B without gap : Let us take A and B as one. 4! = 24 for A and then B let us keep B at 3rd place and A at 1st place =3!there are total 6 such possibilities so we have 6*6 = 36 total possibilities = 60 answer Meljun Cortes 27 5 DECEMBER 2008
28. 28. 5 persons are sitting in a round table in such a way that the tallest person always sits next to the smallest person? Keep tallest and smallest person as 1. we have (4-1)! = 6the tallest and the smallest person can be interchanged = 2 =12 Meljun Cortes 28 5 DECEMBER 2008
29. 29. How many words can be formed from MOBILE so thatconsonent always occupies odd place ? There are 3 odd and 3 even places. We have 3! *3! =36 answerMeljun Cortes 29 5 DECEMBER 2008
30. 30. In how many ways can we arrange 6 + and 4 – signs so that no two – signs are together? ++++++there are 5 places between 2 +. one on extreme left and one on extreme right. We have 7 positions for – sign 7c4 we have 6 places for 6 + sign, so we have 6c6 total = 35 answerMeljun Cortes 30 5 DECEMBER 2008
31. 31. There are 10 buses between Bikaner and Jaipur. In how many ways can Gajendra go to Jaipur and come back without using the same bus in return journey? There are 10 options while goingthere are 9 options while returning (one bus used earlier will not be used) 10*9 = 90 answerMeljun Cortes 31 5 DECEMBER 2008
32. 32. In how many ways can yamini distribute 8 sweets to 8 persons provided the largest sweet is served to Jigyasha? 1 sweet is fixed so we have 7! = 5040 answerMeljun Cortes 32 5 DECEMBER 2008
33. 33. Yamini & Jigyasha go to a train and theyfind 6 vacant seats. In how many ways can they sit?Yamini has 6 options but Jigyasha has only 5 options left = 6*5 = 30 answerMeljun Cortes 33 5 DECEMBER 2008
34. 34. How many words can you make from DOGMATIC? 8! 40320 answerMeljun Cortes 34 5 DECEMBER 2008
35. 35. Gajendra has 12 friends out of whom 8 arerelatives. In how many ways can he invite 7 in such a way that 5 are relatives? 8c5 * 4c2 =56*6 =336 answer Meljun Cortes 35 5 DECEMBER 2008
36. 36. There are 8 points on a plane. No 3points are on a straight line. How many traiangles can be made out of these ? 8c3 = 56 answerMeljun Cortes 36 5 DECEMBER 2008
37. 37. In how many ways can you forma committee of 3 persons out of 12 persons ? 12c3 =220 answerMeljun Cortes 37 5 DECEMBER 2008
38. 38. How many different factors are possible from 75600 ? The factors are : 2^4* 3^3*5^2 *7formula = (number of factors +1) (number of factors +1) .... -1 (4+1)(3+1)(2+1)(1+1) -1 =119 answer Meljun Cortes 38 5 DECEMBER 2008
39. 39. A box contains 7 red 5 white and 4 blue balls. How many selections can be made that we pick up 3 balls and all are red? It is a question of combination. Total possibilities = 7c3 7c3 = 7*6*5 / 3*2*1 = 35 thus there are 35 chances of gettingMeljun Cortes 39 5 DECEMBER 2008
40. 40. A box contains 7 red 5 white and 4 blue balls. What is the probability that in our selections we pick up 3 balls and all are red? Total possibilities for red = 7c3 7c3 = 7*6*5 / 3*2*1 = 35 total possibility of 3 balls : 16c3 =(16*15*14/3*2*1) =560 probability - thus there are 35/560 chances of getting red in all the three selectionsMeljun Cortes 40 5 DECEMBER 2008
41. 41. What is the probability of getting 3 heads when I toss a coin 5 times?This is a case of binomial probability (where there are only 2 outcomes possible, we can use this theory) Here we can use this formula : Ncr (p)^r * (q)^(n-r) =n =5, p = ½ q = (1-p) = ½ , r = 3 5c3 (1/2)^3*(1/2)^2 =5/48 answerMeljun Cortes 41 5 DECEMBER 2008
42. 42. In how many ways can Gajendra invite someor all of his 5 friends in party hosted by him? (at least 1) Frmula of combination of 1 to all = 2^n – 1 = 2^5 - 1 = 32-1 =31 answer Meljun Cortes 42 5 DECEMBER 2008
43. 43. How many words can be formed by using all theletters of the word DRAUGHT so that a. vowels always come together & b. vowels are never together?A There are 2 vowels. We treat them as 1. solution : 6!*2! = 1440 answer b. total possibilities = 7! = 5040 number of cases when vowels are not together = 5040- 1440 = 3600 answerMeljun Cortes 43 5 DECEMBER 2008
44. 44. In how many ways can a cricketeleven be chosen out of a batch of 15 players. 15c11 =15! / ((15-11)!*11!) =15!/(4!*11!) =(15*14*13*12)/(4*3*2*1) 1365 answerMeljun Cortes 44 5 DECEMBER 2008
45. 45. In how many a committee of 5 members can beselected from 6 men 5 ladies consisting of 3 men and 2 ladies 6c3 *5c2 =[(6*5*4)/(3*2*1)] [(5*4)/(2*1)] =20*10 =200 answerMeljun Cortes 45 5 DECEMBER 2008
46. 46. How many 4-letter word with or withoutmeaning can be formed out of the letters of the word LOGARITHMS if repetition of letters is not allowed 10p4 =(10*9*8*7) =5040 answerMeljun Cortes 46 5 DECEMBER 2008
47. 47. how many ways can the letterof word LEADER be arranged We have two e, so divide 6p6 by 2 6!/2! =720 / 2 =360 answerMeljun Cortes 47 5 DECEMBER 2008
48. 48. How many arrangements can be made out of the letters of the word MATHEMATICS be arranged so that the vowels always come together Let us treat all 4 vowels as 1 total digits are 11 we we take 11 – 4+1 = 8 digits vowels can be arranged among themselves = 4!/2! =8!/ (2!*2!) * 4!/2! = 120960 answerMeljun Cortes 48 5 DECEMBER 2008
49. 49. In how many different ways can the letter of the word DETAIL be arranged in such a way that the vowels occupy only the odd positions We have 3 odd and 3 even positions =3! *3! =36 answerMeljun Cortes 49 5 DECEMBER 2008
50. 50. How many 3 digit numbers can be formed fromthe digits 2,3,5,6,7 and 9 which are divisible by 5 and none of the digits is repeated? Last digit must be 5 now we have 5 options for 1st and 4 options for 2nd digit =5*4 = 20 answerMeljun Cortes 50 5 DECEMBER 2008
51. 51. In how many ways can 21 books on English and19 books on Hindi be placed in a row on a self so that two books on Hindi may not be together? We have 22 places for Hindi books. 22p19 *21! Meljun Cortes 51 5 DECEMBER 2008
52. 52. Out of 7 constants and 4 vowels how many words of 3 consonants and 2 vowels can be formed? Selection of 5 digits =7c3 *4c2 =35*6 = 210 5 digits can be arranged in 5! ways =120 total options : 210*120 = 25200 answerMeljun Cortes 52 5 DECEMBER 2008
56. 56. http://www.esnips.com/doc/a4161e7a-6859-4583-b3b9-fc8883819ca2/Rhttp://www.esnips.com/doc/292f3066-dfff-4507-9ef8-f07991c83ca6/reahttp://www.esnips.com/doc/61098141-2b35-4dd2-8c71-a26c481e4406/25http://www.esnips.com/doc/6335a093-fa8b-4829-91b7-99614c3b36b4/1http://www.esnips.com/doc/17998a15-f26a-4de7-b8a7-2b964fcde13f/Lohttp://www.esnips.com/doc/816ec880-c1d5-436c-b5f3-d05525416f0e/1-Ahttp://www.esnips.com/doc/026362a7-efe4-42ef-afe2-1cf7c672d11f/17-rehttp://www.esnips.com/doc/ebceaf38-3af0-4b62-bd4a-1a9c1b002d47/re Meljun Cortes 56 5 DECEMBER 2008