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Combinations

Learn the concepts of permutations and combinations in mathematics, including formulas and examples. Understand how to count subsets and sequences effectively. Explore the multiplication principle and the theorem on combinations.

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Combinations

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  1. Combinations • We should use permutation where order matters • We should use combination where order does not matter. • Combination: An r-element subset of an n-element set A is a combination of A, taken r at a time. Example: A = {1, 2, 3, 4} Distinct combinations of A, taken three at a time: (different subsets of A, each with r elements) A1 = {1, 2, 3], A2 = {1, 2, 4}, A3 = {1, 3, 4}, A4 = {2, 3, 4} Note: A1 = {1, 2, 3} = {2, 3, 1} = {3, 1, 2} = {3, 2, 1} = {2, 1, 3} These are subsets where order is not important. These are not sequence where order is important.

  2. Combinations • How to count the number of r-element subsets of an n-element set A? • Each permutation of the elements of A, taken r at a time, can be produced by doing the following tasks in sequence. • Task 1: Choose a subset B of A containing r elements • Task 2: Choose a particular permutation of B Suppose, C => No. of ways to choose B Task2 can be performed in r! ways. Total no. of ways of performing both task is (using multiplication principle) = C. r! Hence, C. r! = n P r or, C = nPr / r! = n! /r! (n – r)!

  3. Combinations • The number of combinations of the elements of A, taken r at a time (Number of r-element subsets of A) is given by nCr = n! r! (n – r)!

  4. Note: Counting • In general, • When order matters, we count the number of sequences or permutations. • When order does not matter, we count the number of subsets or combinations.

  5. Theorem on combinations Suppose k selections are to be made from n items without regard to order and that repeats are allowed, assuming at least k copies of each of the n items. The number of ways these selections can be made is (n+k-1) C k.

  6. Example • Suppose that a valid computer password consists of seven characters, the first of which is a letter chosen from the set {A, B, C, D, F, G} and the remaining six characters are letters chosen from the English alphabet or a digit. How many different passwords are possible? • We need to perform two tasks: Task 1: Choose a starting letter from the set given. Task 2: Choose a sequence of letter and digits. Repeats are allowed. Task 1 can be performed in 7C1 = 7 ways. Since there are 26 letters and 10 digits, task 2 can be performed in 36 6 = 2,176,782,336 ways. By using multiplication principle, the number of different passwords are 7. 2,176,782,336 = 15,237,476,352.

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