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Permutations: Arrangements of Distinct Objects A permutation of n (distinct objects) is an arrangement of all of the objects in a definite order. The total number of such permutations is denoted by Note that, by the Fundamental Counting Principle, Example 1: A choir has learned eight songs for its spring concert. In how many different ways can the director arrange theses songs to form the concert program? Example 2: An investment club with five members wants to select a president and a vice-president. In how many ways can this be done? Suppose you have n objects to choose from, but only want to select some rather than all of them. Def.: A permutation of n (distinct) objects taken r at a time is an arrangement of r of the n objects in a definite order. The total number of such objects is denoted by or Proof: Suppose we want to form an arrangement of r objects from a set of n objects. 1 2 3 4 5 r-2 r-1 Thus, by the Fundamental Counting Principal, Ex 3: Evaluate each expression. a) P(8,3) b) P(6,6) r Ex 4: In how many ways can a chairman, treasurer, and a secretary be selected from a Board of Directors with eight members? Ex 5: a) Find the number of permutations of the letters of the word DIPLOMA for which the letter L remains in the middle position. b) How many ways are there of arranging the letters of DIPLOMA so that the letters O and I are together? Ex 6: At a used car lot, six cars for sale are to be parked side-by-side. In how many different ways can this be done if: a) the one (and only one) car with sunroof must be at the right end of the line? b) the three black cars must be together? Homework: Pg. 255: #4-6, 8 a,b iii, iv