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Permutations & Combinations

Permutations & Combinations. Section 2.4. Permutations. When more than one item is selected (without replacement) from a single category and the order of selection is important , the various possible outcomes are called permutations . The notation used for permutations is n P r .

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Permutations & Combinations

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  1. Permutations & Combinations Section 2.4

  2. Permutations • When more than one item is selected (without replacement) from a single category and the order of selection is important, the various possible outcomes are called permutations. • The notation used for permutations is nPr. • The number of permutations of r items selected without replacement from a pool of n items (note r ≤ n) is given by: nPr =

  3. Example 1: Choosing a Committee of 4 members where order matters • The Board of Directors of XYZ Corporation has 15 members. In how many ways can we choose a committee of 4 members where we have a President, Vice-President, Treasurer and Secretary? • Here we have a situation where order matters. • The pool of candidates is 15, that is n = 15. • The number of seats on the committee is 4, hence r = 4. • The number of committees would be:

  4. Combinations • When one or more item is selected (without replacement) from a single category and the order of selection is not important, the various outcomes are called combinations. • The notation used for combinations is . • The number of combinations of r items selected without replacement from a pool of n items (note r ≤ n) is given by:

  5. Example 2: Choosing a Committee of 4 members where order does not matter • The Board of Directors of XYZ Corporation has 15 members. In how many ways can we choose a committee of 4 members, (each member has equal rank)? • Here order does not matter. • The pool of candidates is 15, that is n = 15. • The number of seats on the committee is 4, hence r = 4. • The number of committees would be:

  6. Permutations of identical items • Say you want all the distinct permutations of the word SEE. How many do you have? • Colorize the two Es. Then list the permuations SEE, SEE, ESE, ESE, EES, EES • If you take away the color you get repeats: SEE, SEE, ESE, ESE, EES, EES • The only distinct permutations are SEE, ESE, EES, i.e. 3. • The formula for the number of distinct permuations is • Where n is the total number of items and x is the number of times the first item is repeated, y is the number of times the second item is repeated, z is the number of times the third item is repeated etc. • Example: SEE, n = 3, (Number of S) x = 1, and (Number of E) y = 2. • Example: How many distinct permutations are there for the word MISSISSIPPI? • ANSWER: n = 11, w = 1(M), x = 4 (I), y = 4 (S), z = 2 (P):

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