Assignment 77

 "If a lunatic scribbles a jumble of mathematical symbols it does not follow that the writing means anything merely because to the inexpert eye it is indistinguishable from higher mathematics" - (Eric Temple Bell, 1883-1960)   Mary Dolciani Halloran (1923-1985): A great American mathematician, Mary Dolciani had a considerable impact on modern mathematics education. An inspiring teacher at Hunter College, she authored over thirty mathematics books. A specialist in number theory and modern algebra, she was active in many professional organizations and lectured to teachers and administrators throughout the United States. Why does Herkimer think that spooks have very high phone bills? Answer: They make lots of ghost-to-ghost calls. Herky wants to know: Why is it there there are people who can cook but don't, and people who can't cook but do? Lots of people think they would be happy if they had all the money they wanted. Would they also be happy if they had all the money their creditors wanted? ASSIGNMENT #77 Reading: Section 12.3, pages 716-719. Written: Page 719/12-23. (You can write answers neatly in your text.) Page 757/15-18 (Be able to do binomial expansions.)
 Mathematical numbers:ABUNDANT NUMBER: An integer number is abundant if the sum of its proper divisors (all divisors except the number itself) is greater than the number. The proper divisors of 12 are 1,2,3,4,6. Since 1+2+3+4+6 = 16, which is greater than 12, the number 12 is an abundant number. Other abundant numbers include 18, 20, 24, 30, and 36.
OK, just some interesting numbers from the amazing world of permutations and combinations...

The number of ways that 15 people could arrange themselves in a line is 15! = 15(14)(13)(12)(11)(10)(9)(8)(7)(6)(5)(4)(3)(2)(1). This is approximately 1,308,000,000,000, or about 1.3x1012. If these people could possibly demonstrate one arrangement per second, it would take them 41,466 years, or over 414 centuries, to demonstrate all possible arrangements.

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A bridge hand consists of 13 cards dealt from a deck of 52 cards. The number of possible bridge hands is 52C13 , which is approximately 6.35x1011 = 635,000,000,000. If you could demonstrate one bridge hand per second, it would take you 20,136 years, or over 201 centuries, to show all possible bridge hands.

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On the other hand, if we consider 5-card hands (many forms of poker involve 5-card hands), the number of possible poker hands is 52C5 = 2,598,960. If you could demonstrate one such poker hand per second, it would take you slightly over 30 days to demonstrate all possible 5 card poker hands.

If you had a 5-card poker hand containing all hearts, this is called a flush (and is generally considered to be a good hand). The probability that you would be dealt a 5-card flush is (13C5)/(52C5) = (1,287)/(2,598,960) = 4.95x10-4, or about 0.0005 =0.05%. In other words, if you were continuously dealt 5 cards from a thoroughly shuffled deck of 52 cards, you would expect to get five hearts once every 2,000 hands.

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Oh, the power of mathematics!

 Problem: What is the probability that randomly constructed bridge hand of 13 cards contains no hearts? Solution (with communication): The hand would have to be constructed from the 39 non-hearts in the deck. The requested probability is (39C13)/(52C13) = 0.01279, or about 1.3%. Problem: A mathematics class contains 12 girls and 8 boys. If four students are randomly chosen from this class, (a) what is the probability that all of the students are girls? (b) what is the probability that at least one of the chosen students is a boy? Solution (with communication): (a) Probability (all girls) = (12C4)/(20C4) = 0.102167, or about 10.2%. (b) Probability (at least one boy) = 1 - probability (all girls) = 1 - 0.102167, or about 89.8%.