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Show that there are 10 distinct ways to distribute

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Unformatted text preview: secret ballot such that the voter’s identities are kept confidential. 1 2 3 Show that there are 45 distinct ways to distribute the eight votes among the three candidates. Show that there are 10 distinct ways to distribute the eight votes among the three candidates so that A will win the election. How many distinct ways are there to distribute the eight votes among the three candidates so that nobody will win the election? Solution to Problem 4 I Solution. 1 This is a problem of partitioning n indistinguishable objects (8 votes) into r distinguishable groups (3 candidates). 8+3−1 8 = = 2 10! 8! × 2! 45 As we have fixed the result to be A winning the election, we first allocate 5 votes to A and consider the number of ways to distribute the remaining 3 votes. The problem becomes a partition of n indistinguishable objects (3 votes) into r distinguishable groups (3 candidates). 3+3−1 3 = = 5! 3! × 2! 10 Solution to Problem 4 II 3 We can simply deduct the number of ways to distribute the votes for candidate A, B and C to win respectively from the total number of ways to distribute the eight votes. This is based on the fact that the events {A winning}, {B winning}, {C winning} and {nobody winning} are mutually exclusive and exhaustive. 45 − 10 − 10 − 10 = 15 Problem 5 Problem 5. A total of r identical robots are being lifted from the ground floor to the n upper floors of a building. 1 2 Find the total number of distinct patterns. A student claims that if each robot is independently and equally probably allocated to each of the n floors, then the probability that all robots end up on the first floor is given by Number of ways of sending all robots to 1/F = Number of distinct exit patterns Comment on the above statement. Is it correct? n+r −1 r −1 . Solution to Problem 5 Solution. 1 This is a problem of allocating r identical objects into n distinct containers. Thus the total number of ways is r +n−1 r 2 . The statement is wrong because, if each robot is independently and equally...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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