A1-sol - Stat 230 Assignment 1 Solutions The first three...

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Stat 230 Assignment 1 - Solutions The first three questions consider the process of arranging coloured marbles in a row from left to right. Two marbles of the same colour are to be considered indistinguishable when counting arrangements. 1. (a) Suppose that there are 4 white and 2 black marbles. One way to arrange the marbles is . List all 15 ways to arrange these marbles. (b) What combinatorial number describes the number of ways to arrange n black and 4 white marbles? Check your answer for the case n = 2 . To pick an arrangement, choose 4 out of n + 4 positions to hold black marbles. This can be done in ( n +4 4 ) ways. In the case of n = 2 , this corresponds to the ( 6 4 ) = 6 · 5 · 4 · 3 4! = 15 arrangements listed in part (a). (c) In which of the arrangements in 1a (list them) is every white marble adjacent to at least one other white marble? (d) List the ways to arrange 2 black and 2 red marbles. (e) Explain why the lists in 1c and 1d have the same size? (Hint: Consider replacing each red marble with two consecutive white marbles.) If 4 white marbles are arranged so that each is adjacent to at least one other, then they must appear either in two groups of 2 or in a single group of 4 . In both cases, the white marbles appear in even groups and each group may be replaced with half as many red marbles. The process may be reversed by replacing each red marble with two adjacent white marbles. (f) How many ways are there to arrange n black and 4 white marbles so that every white marble is adjacent to at least one other white marble? (Check that your answer agrees with part 1c.) Using the logic from part (e), the required arrangements may be counted by counting, instead, all arrangements of n black and 2 red marbles, of which there are ( n +2 2 ) (using the logic from (b)) . This agrees with part (c), since ( 2+2 2 ) = 4 · 3 2! = 6 . (g) If 22 black and 4 white marbles are arranged at random, what is the probability that every white marble is adjacent to at least one other white marble? By part (f), there are ( 22+2 2 ) arrangements of the marbles with each white ball ad- jacent to another, out of ( 22+4 4 ) equally likely arrangements total, so the required probability is ( 24 2 ) ( 26 4 ) = 6 325 0 . 01846 ... . (Page 1 of 7)
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Stat 230 Assignment 1 - Solutions 2. (a) Suppose that there are 2 black and 5 white marbles. One way to arrange the marbles is . List all 21 ways to arrange these marbles. (b) What combinatorial number describes the number of ways to arrange n black and 5 white marbles? Check your answer for the case n = 2 . As in 1(b), there are ( n +4 4 ) ways to choose the arrangements. In the case of n = 2 , this corresponds to the ( 7 5 ) = 7 · 6 · 5 · 4 · 3 5! = 21 arrangements listed in part (a). (c) In which of the arrangements in 2a (list them) is every white marble adjacent to at least one other white marble? (Note: There are nine such arrangements.) (d) i. List the ways to arrange 2 black, 1 red, and 1 blue marble.
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