Last updated at March 8, 2017 by Teachoo

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Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, Finding total number of arrangements In word INDEPENDENCE There are 3N, 4E, & 2D, 1I, 1P & 1C Since letters are repeating so we use this formula ๐!๏ทฎ๐1!๐2!๐3!๏ทฏ Total letters = 12 So, n = 12 Since, 3N, 4E, & 2D p1 = 3, p2 = 4,p3 = 2 Total arrangements = 12!๏ทฎ3!4!2!๏ทฏ = 1663200 Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, โข do the words start with P If the word start with P We need to arrange (12 โ 1) = 11 We need to arrange letters I,N,D,E,E,N,D,E,N,C,E Here, 4E, 3N,2D Since letters are repeating since we use this formula Number of arrangements = ๐!๏ทฎ๐1!๐2!๐3!๏ทฏ Total letters to arrange = 11 So, n = 11 Since, 4E, 3N,2D p1 = 4 , p2 = 3 , p3 = 2 Number of arrangements = ๐!๏ทฎ๐1!๐2!๐3!๏ทฏ = 11!๏ทฎ4!3!2!๏ทฏ = 138600 Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, do the words start with P If the word start with P We need to arrange (12 โ 1) = 11 We need to arrange letters I,N,D,E,E,N,D,E,N,C,E Here, 4E, 3N,2D Since letters are repeating since we use this formula Number of arrangements = ๐!/๐1!๐2!๐3! Total letters to arrange = 11 So, n = 11 Since, 4E, 3N,2D p1 = 4 , p2 = 3 , p3 = 2 Number of arrangements = ๐!/๐1!๐2!๐3! = 11!/4!3!2! = 138600 Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (ii) do all the vowels always occur together There are 5 vowels in the given word โINDEPENDENCEโ i.e. 4Eโs & Iโs They have occur together we treat them as single object we treat as a single object So our letters become We arrange them now Hence the required number of arrangement = 8!/3!2! ร 5!/4! = ((8 ร 7 ร 6 ร 5 ร 4 ร 3!) ร(5 ร 4!))/(3! ร 4! ร 2) = 16800 Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (iii) do the vowels never occur together Number of arrangements where vowel never occur together = Total number of arrangement โ Number of arrangements when all the vowels occur together = 1663200 โ 16800 = 1646400 Example 16 Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (iv) do the words begin with I and end in P? Lets fix I and P at Extreme ends Since letters are repeating, Hence we using this formula ๐!/๐1!๐2!๐3! Here Total letters = n = 10 Since 2D, 4E, 3N p1 = 2, p2 = 4, p3 = 3 Required number of arrangement = 10!/2!4!3! = 12600

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Chapter 7 Class 11 Permutations and Combinations

Serial order wise

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.