Permutations & CombinationsPermutationsExample 1: Choosing a Committee of 4 members where order mattersCombinationsExample 2: Choosing a Committee of 4 members where order does not matterPermutations of identical itemsPermutations & CombinationsSection 2.4Permutations•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 = )!(!rnnExample 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:7 60,3 2121 31 41 51234567891011123456789101112131415!11!15)!41 5(!15415PCombinations•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:rnC!)!(!rrnnCrnExample 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:13 652432760123412131415)1234)(1234567891011(123456789101112131415!4!11!15!4)!415(!15rnCPermutations 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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