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Unformatted text preview: 11COMP170 – Fall 2007Midterm 1 Review21Question 1nguests are arranged seats in a row facing the audiencea) How many different ways are there to seat the n guestsat n seats?22Question 1Note that the order is important since the guests could belisted from left to right from the audience’s perspectiven!nguests are arranged seats in a row facing the audiencea) How many different ways are there to seat the n guestsat n seats?23Question 1nguests are arranged seats in a row facing the audienceb) Letn >2. How many ways to seat n guests if twospecific guests will not sit next to each other?24Question 1nguests are arranged seats in a row facing the audienceb) Letn >2. How many ways to seat n guests if twospecific guests will not sit next to each other?# waysnot sitting together=n!# ways sitting together25Question 1nguests are arranged seats in a row facing the audienceb) Letn >2. How many ways to seat n guests if twospecific guests will not sit next to each other?# waysnot sitting together=n!# ways sitting togetherTreating the two people as one indivisible group, there are(n1)!different ways of seating the guests.Withinthe twoperson group, there are 2 ways to sit them.# ways sitting together =2(n1)!# waysnotsitting together =n!2(n1)!=(n2)(n1)!26Question 1nguests are arranged seats in a row facing the audiencec)n >10. Guests include 5 couples. For each couple,husband must sit with wife. How many seatings are there?27Question 1nguests are arranged seats in a row facing the audiencec)n >10. Guests include 5 couples. For each couple,husband must sit with wife. How many seatings are there?Treat each couple as an indivisible group, and others asindividual groups.There are(n5)!ways to seatn5groups.For each couple (group),there are 2 ways to seat husband and wife.# ways to seat guests =25(n5)!31Question 2Zn={, ..., n1}a) How many 5element subsets ofZ10contain at least oneelement inZ3?32Question 2# 5element sets ofZ10=(105)# 5element sets ofZ10notcontaining elements ofZ3=(75)# 5element sets ofZ10containingat least oneelement ofZ3Zn={, ..., n1}a) How many 5element subsets ofZ10contain at least oneelement inZ3?=1057533Question 2b) How many 5element subsets ofZ10contain two odd andthree even numbers?34Question 2b) How many 5element subsets ofZ10contain two odd andthree even numbers?Z10={,1,2,3,4,5,6,7,8,9}(52)ways to choose 2 numbers from{1,3,5,7,9}(53)ways to choose 3 numbers from{,2,4,6,8}By the product principle, we have(52)·(53)35Question 2c) Let n be positive and odd. Show that # of evensizedsubsets ofZnequals # of oddsized subsets ofZn36Question 2c) Let n be positive and odd. Show that # of evensizedsubsets ofZnequals # of oddsized subsets ofZnLetOn={1,3, .5, ..., n}Since(nk)=(nnk), we haveXk∈Onnk=Xk∈Onnnk=Xk∈EnnkEn={,2,4, ..., n1}f(k) =nkdefines a bijection betweenOnandEn37Question 2c) Let n be positive and odd. Show that # of evensizedsubsets ofZnequals # of oddsized subsets ofZnAlternatively:2n= (1 + 1)n=X≤i≤nni!...
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 Spring '10
 M.J.Golin
 Computer Science, #, Quantification, Existential quantification, 4 digits, Z12

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