Permutation & Combination — Set #3 | Practice

{"setId":"478bbcf0-d59d-46ba-a6ae-16c88e88a28c","setTitle":"Permutation and Combination 3 – UPSC CSAT","topicName":"Permutation & Combination","topicSlug":"permutation-combination","setNumber":3,"totalQuestions":10,"questions":[{"id":"008733fc-87ad-4353-9619-3acc853c70ed","order":1,"statement":"What are the number of possible 3-digit numbers that can be formed using the digits 0, 1, 3, 5 and 8 (repetition of digits is allowed)?","options":{"A":"32","B":"64","C":"100","D":"80"},"correctOption":"C","solution":"In a 3-digit number, first place can be occupied in 4 ways (1, 3, 5 or 8, not 0). Second and third place can each be occupied in 5 ways (0, 1, 3, 5 or 8). Number of 3-digit numbers = 4x5x5 = 100. Hence, option (c) is correct."},{"id":"942cd70b-c085-4f70-9930-962947868b76","order":2,"statement":"Five coins are tossed simultaneously. In how many of the outcomes will the third coin turn up a head?","options":{"A":"2^5","B":"2^4","C":"2^6","D":"2^7"},"correctOption":"B","solution":"When 5 coins are tossed, total outcomes = 2^5. If the third coin must be heads (fixed), the remaining 4 coins have 2^4 outcomes. So the number of outcomes = 1 x 2^4 = 2^4. Hence, option (b) is correct."},{"id":"2b2528f2-3211-4600-9ced-e807c0ebeb7d","order":3,"statement":"In how many ways can Rohan send wedding invites to 7 of his friends, if he has 5 different modes of communications to do so?","options":{"A":"35","B":"7^5","C":"5^7","D":"None of these"},"correctOption":"C","solution":"Rohan has 5 ways to send invites for each of the 7 friends independently. So the number of ways = 5x5x5x5x5x5x5 = 5^7. Hence, option (c) is correct."},{"id":"82f61f74-eef1-4151-83f1-885ff12902ef","order":4,"statement":"In how many ways can the letters of the word ANSWER be rearranged such that the vowels always appear together?","options":{"A":"6!","B":"5! x 2!","C":"4! x 2!","D":"3! x 2!"},"correctOption":"B","solution":"ANSWER has 6 letters with 2 vowels (A, E). Grouping the vowels as one unit gives 5 elements, arranged in 5! ways. The 2 vowels can be arranged among themselves in 2! ways. Total = 5! x 2!. Hence, option (b) is correct."},{"id":"90d1df38-f5a1-4fcd-8a19-2d9d088d538b","order":5,"statement":"Find the number of ways in which 5 letters can be posted in three post boxes, if any number of letters can be posted in any of the three post boxes.","options":{"A":"3^5","B":"5^3","C":"5C3","D":"5P3"},"correctOption":"A","solution":"Each of the 5 letters independently has 3 choices of post box. Total ways = 3x3x3x3x3 = 3^5. Hence, option (a) is correct."},{"id":"8575e9f7-fd24-4397-b75f-16e59ff46be1","order":6,"statement":"Vivek's Home, Market and office are located in such a manner that there are 3 different roads from Home to market, 4 different roads from Market to office, and 3 different roads directly from Home to office. In how many different ways can Vivek travel from Home to office using these roads?","options":{"A":"11","B":"9","C":"12","D":"15"},"correctOption":"D","solution":"Direct ways from Home to office = 3. Ways via Market = 3 (Home to Market) x 4 (Market to office) = 12. Total ways = 3 + 12 = 15. Hence, option (d) is correct."},{"id":"4fb7c8cd-8873-4766-9c4e-917f5d7187ec","order":7,"statement":"How many odd 4-digit positive integers that are multiples of 5, can be formed without using the digit 4?","options":{"A":"648","B":"729","C":"1296","D":"916"},"correctOption":"A","solution":"For an odd multiple of 5, the unit digit must be 5 (0 would make it even). Leftmost digit cannot be 0 or 4, giving 8 options. Hundreds and tens places can be 0-9 except 4, giving 9 options each. Total = 8x9x9x1 = 648. Hence, option (a) is correct."},{"id":"014f790c-2148-4fb3-a7ac-43885e8f70aa","order":8,"statement":"In how many ways can the letters of the word 'CABINET' be rearranged to make 7 letter words such that none of the letters repeat?","options":{"A":"49","B":"7^7","C":"7!","D":"7C7"},"correctOption":"C","solution":"CABINET has 7 distinct letters. The first position can be filled in 7 ways, the second in 6 ways, and so on. Total ways = 7x6x5x4x3x2x1 = 7!. Hence, option (c) is correct."},{"id":"2de89c02-2ce6-4f9c-b431-8e368314596d","order":9,"statement":"Find the total number of 4-digit numbers having at least one of their digits 4?","options":{"A":"2126","B":"3168","C":"2968","D":"2624"},"correctOption":"B","solution":"Total 4-digit numbers (1000-9999) = 9x10x10x10 = 9000. 4-digit numbers with no digit as 4 = 8x9x9x9 = 5832. Numbers with at least one digit as 4 = 9000 - 5832 = 3168. Hence, option (b) is correct."},{"id":"66eb4588-95ef-468d-ac90-297d184f81d7","order":10,"statement":"In a question paper there are seven questions to be attempted. Each question can only be answered as either - True (T) or False (F). Each candidate answered all the questions. No two candidates have given the answers to the seven questions in an identical sequence. For this to happen the maximum number of candidates is:","options":{"A":"14","B":"64","C":"1028","D":"128"},"correctOption":"D","solution":"Each of the 7 questions can be answered in 2 ways, so total distinct sequences = 2x2x2x2x2x2x2 = 2^7 = 128. If a 129th candidate is selected, the sequence would repeat. Hence, option (d) is correct."}]}