MIT6_042JS10_assn09

# MIT6_042JS10_assn09 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science April 7 Prof. Albert R. Meyer revised April 15, 2010, 1374 minutes Problem Set 9 Due : April 16 Reading: Notes Ch. 16.1 16.9 Problem 1. Show that for any set of 201 positive integers less than 300, there must be two whose quotient is a power of three (with no remainder). Problem 2. Answer the following questions with a number or a simple formula involving factorials and bino- mial coefﬁcients. Brieﬂy explain your answers. (a) How many ways are there to order the 26 letters of the alphabet so that no two of the vowels a , e , i , o , u appear consecutively and the last letter in the ordering is not a vowel? Hint: Every vowel appears to the left of a consonant. (b) How many ways are there to order the 26 letters of the alphabet so that there are at least two consonants immediately following each vowel? (c) In how many different ways can 2 n students be paired up? (d) Two n -digit sequences of digits 0,1,. . . ,9 are said to be of the same type if the digits of one are a permutation of the digits of the other. For n = 8 , for example, the sequences 03088929 and 00238899 are the same type. How many types of n -digit integers are there? Problem 3. Section 16.8.3 explained why it is not possible to perform a four-card variant of the hidden-card magic trick with one card hidden. But the Magician and her Assistant are determined to ﬁnd a way to make a trick like this work. They decide to change the rules slightly: instead of the Assistant lining up the three unhidden cards for the Magician to see, he will line up all four cards with one card face down and the other three visible. We’ll call this the face-down four-card trick . For example, suppose the audience members had selected the cards

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## This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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MIT6_042JS10_assn09 - Massachusetts Institute of Technology...

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