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Unformatted text preview: 4.3. GENERALIZED PERMUTATIONS AND COMBINATIONS 67 4.3. Generalized Permutations and Combinations 4.3.1. Permutations with Repeated Elements. Assume that we have an alphabet with k letters and we want to write all possible words containing n 1 times the first letter of the alphabet, n 2 times the second letter,. . . , n k times the k th letter. How many words can we write? We call this number P ( n ; n 1 , n 2 , . . . , n k ), where n = n 1 + n 2 + + n k . Example : With 3 a s and 2 b s we can write the following 5-letter words: aaabb , aabab , abaab , baaab , aabba , ababa , baaba , abbaa , babaa , bbaaa . We may solve this problem in the following way, as illustrated with the example above. Let us distinguish the different copies of a letter with subscripts: a 1 a 2 a 3 b 1 b 2 . Next, generate each permutation of this five elements by choosing 1) the position of each kind of letter, then 2) the subscripts to place on the 3 a s, then 3) these subscripts to place on the 2 b s. Task 1) can be performed in P (5; 3 , 2) ways, task 2) can be performed in 3! ways, task 3) can be performed in 2!. By the product rule we have 5! = P (5; 3 , 2) 3! 2!, hence P (5; 3 , 2) = 5! / 3! 2!. In general the formula is: P ( n ; n 1 , n 2 , . . . , n k ) = n ! n 1 ! n 2 ! . . . n k ! . 4.3.2. Combinations with Repetition. Assume that we have a set A with n elements. Any selection of r objects from A , where each object can be selected more than once, is called a combination of n objects taken r at a time with repetition . For instance, the combinations of the letters a, b, c, d taken 3 at a time with repetition are: aaa , aab , aac , aad , abb , abc , abd , acc , acd , add , bbb , bbc , bbd , bcc , bcd , bdd , ccc , ccd , cdd , ddd . Two combinations with repetition are considered identical...
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