# Worksheet: Permutations and Combinations (solutions)

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V63.0233: Theory of ProbabilitySolutionsWorksheet for Sections 1.3–1.4 : Permutations and CombinationsJune 30, 20091.Among the 16 applicants for four diﬀerent teaching positions in an elementary school, onlyten have master’s degrees.(i) In how many ways can these positions be ﬁlled?(ii) In how many ways can these positions be ﬁlled with applicants having master’s degrees?(iii) If one of the positions requires a master’s degree, while for the others it’s optional, in howmany ways can the four positions be ﬁlled?Solution.(i) Notice that the positions are described as diﬀerent so order matters. We have 16ways to ﬁll the ﬁrst spot, 15 to ﬁll the second, etc., until you ﬁll four spots:16 × 15 × 14 × 13 = 43, 680(ii) We have 10 ways to ﬁll the ﬁrst, spot, 9 ways to ﬁll the second, and so on:10 × 9 × 8 × 7 = 5040(iii)2.How many anagrams of the word MASSACHUSETTS can you ﬁnd?Solution. Of the 13 letters, we have two A’s, four S’s, and two T’s. This gives a total of13!= 64, 864, 8004!2!2!anagrams.13.A student takes a true-false test of 15 questions. In how many diﬀerent ways can he or shemark this test and get(i) three right and 12 wrong?(ii) six right and nine wrong?(iii) 12 right and three wrong?Solution.(i) We can choose the incorrect three to be any of the questions numbered 1 through15. Hence there are( )15= 4553diﬀerent ways to do this.(ii)( )15= 50056(iii)( )15= 45512(Notice there are exactly the same number of ways to get three right as there are to get threewrong.)4.Rework the MASSACHUSETTS problem by choosing the slots for each of the letters. Forinstance, there are 13 positions for the M. Of the 12 remaining, we need to choose two for the Ts.Of the 10 remaining, we need to choose four for the Ss, etc.Solution. Following the directions, we can choose any of 13 positions for the M. The 2 As need to( )12go in any of the remaining 12 slots; there areways to do this. And so on. We get2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )13121065432= 64, 864, 80012411112MASCHUETThis is the same thing we get in Problem 2.2