Aptitude Training - PERMUTATIONS AND COMBINATIONS 2
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The document discusses the concepts of permutations and combinations, detailing how to arrange or select items from a set of n different elements. It includes examples and formulas for calculating arrangements when certain conditions are met, such as keeping vowels together or arranging groups alternately. Additionally, it provides practice problems for further understanding of the material.
Aptitude Training - PERMUTATIONS AND COMBINATIONS 2
- 2. Basics : Permutation – arrangement Combination – selection • Out of n different things, if some or all the things are selected at a time, then each of these selections is called a Combination • Out of n different things, if r things are selected at a time, then the number of selections is given by = • and = • + + + + ….. • The number of ways in which we can select is given by .
- 3. Permutation – arrangement Combination – selection • Out of n different things, if some or all the things are selected at a time, and arranged, then each of these arrangement is called a Permutation. • N different things are selected all at a time can be arranged in n factorial ways. • Out of n different things, if r things are selected at a time and arranged, then the number of arrangements are given by = Basics :
- 4. 1. In how many ways can u arrange the letters of the word “SUCCESS”, such that all vowels all together? 2. In how many ways can u arrange 4 boys and 4 girls around a circular table such that, no 2 girls sit together? 3. How many 3 digit numbers can be formed using the digits 1,2,3,4,5? 4. How many 4 digit numbers can be formed using the digits 0,1,2,3,4,5 which are divisible by 4 (repetition of digits is allowed)? Permutation :
- 5. In how many ways can u arrange the letters of the word “SUCCESS”? Ans : NOTE: The number of ways in which we can arrange n things such that p things are alike, q things are alike, r things alike is given by . In the given question, C – 2, S – 3, E – 1, U – 1 => = 420 1. number of ways = 420 S UC ECS S
- 6. In how many ways can u arrange the letters of the word “SUCCESS”, such that all ‘S’ are together? Ans : NOTE: The number of ways in which we can arrange n things such that p things are alike, q things are alike, r things alike is given by . In the given question, C – 2, S – 3, E – 1, U – 1 = 60 1a. number of ways = 60 SSS UC EC
- 7. In how many ways can u arrange the letters of the word “SUCCESS”, such that all vowels are together? Ans : NOTE: The number of ways in which we can arrange n things such that p things are alike, q things are alike, r things alike is given by . In the given question, C – 2, S – 3, E – 1, U – 1 => = 120 1b. number of ways = 120 S UEC CS S
- 8. In how many ways can u arrange 4 boys and 4 girls around a circular table such that all boys sit together? 2. Number of ways are = 576 NOTE : N different things can be arranges around circular table in (N-1)! Ways G1G2G3G4 4! 5-1) ! B1B2B3B4
- 9. In how many ways can u arrange 4 boys and 4 girls around a circular table such that no two girls sit together? 2a. Number of ways are = 144 NOTE : N different things can be arranges around circular table in (N-1)! Ways 3! 5-1) ! 3! ! = 144
- 10. How many 3 digit numbers can be formed using the digits 1,2,3,4,5?3. Ans : _ _ _ => = 3! Number that are formed by 3 digit numbers are = 60 How many 4 digit numbers can be formed using the digits 0,1,2,3,4,5?3a. Ans : _ _ _ _ 5 5 4 3 => 5 4 3 Number that are formed by 4 digit numbers are = 300
- 11. How many 4 digit numbers can be formed using the digits 0,1,2,3,4,5 which are divisible by 4(repetition of digits is allowed)? 4. Ans : Condition: last 2 digits should be divisible by 4 or the number should end with 2 zeroes. _ _ _ _ 00,04,12,20,24,32,40,44,52 5 6 9 => 5 9 = 270 Number that are formed by 4 digit numbers are = 270
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- 13. Practice problems on Permutations & Combinations
- 14. 1. In how many ways can u arrange the letters of the word ___ such that no two vowels are together, all consonants together, all vowels together, no two consonants together? 1. “APTITUDE” 2. “TRIANGLE” 2. In how many ways can u arrange the letters of the word “SPREAD” such that ALL vowels are together? 3. How many 4 digit numbers can be formed using the digits 0,1,2,3,4,5 which are divisible by 4(repetition of digits is not allowed)?
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