Assignment 76
"For the things of this world cannot be made known without a knowledge of mathematics."  (Roger Bacon, 1267)
John von Neumann (19031957): Born in Hungary, von Neumann proved a number of theorems relating to measure theory, quantum theory, and the theory of games while still in his twenties. He later taught at Princeton, where he contributed to the development of lattice theory, continuous groups, shockwave theory, and computer technology. During the Cold War, his ideas were used by both capitalists and communists sides to project minimum strategy losses. Mathematics suffered a great loss at his death as he had much unfinished research.
Von Neumann had a great sense of humor. Upon emerging from his wrecked automobile after a crash, he said: "The trees on the right were passing me in orderly fashion at sixty miles per hour. Suddenly one of them stepped out in my path." 
Why did Herkimer take a lion into the desert just before Christmas?
Answer: He wanted to see sandy claws.
Herky wants to know:
Would cannibals get angry if told their jokes aren't in good taste?
Does the phrase "cold cash" refer to money that doesn't stay in your pocket long enough to get warm? 
ASSIGNMENT #76
Reading: Section 12.2, pages 708711.
Written: Pages 712713/2630, 3638, 4750. 
Mathematical word analysis: CATENARY : From the Latin catenareus (chain). In mathematics, catenary is the name for the shape assumed by a perfectly flexible chord hanging freely between two points of support. Note that in real life, a chain if often used in this manner.  Pascal's Triangle:




1 









_{0}C_{0} 







1 

1 







_{1}C_{0} 

_{1}C_{1} 





1 

2 

1 





_{2}C_{0} 

_{2}C_{1} 

_{2}C_{2} 



1 

3 

3 

1 



_{3}C_{0} 

_{3}C_{1} 

_{3}C_{2} 

_{3}C_{3} 

1 

4 

6 

4 

1 

_{4}C_{0} 

_{4}C_{1} 

_{4}C_{2} 

_{4}C_{3} 

_{4}C_{4} 
Note that (x + y)^{3} = 1x^{3} + 3x^{2}y + 3xy^{2} + 1y^{3}.
Also, (x  2y)^{4} = 1x^{4} + 4x^{3}(2y) + 6x^{2}(2y)^{2} + 4(x)(2y)^{3} + 1(2y)^{4}
= x^{4}  8x^{3}y + 24x^{2}y^{2}  32xy^{3} + 16y^{4}.
OK, here are problems involving combinations. Suppose a political club contains 10 Republicans, 12 Democrats and 5 Independents.
a) How many groups (committees) of five members can be chosen from the club?
Answer: _{27}C_{5} = 80,730.
b) How many groups (committees) of five would contain 2 Republicans, 2 Democrats, and 1 Independent?
Answer: (_{10}C_{2})(_{12}C_{2})(_{5}C_{1}) = (45)(66)(5) = 14,850.
c) If a committee of five is chosen by a random process, what is the probability that the committee contains 2 Republicans, 2 Democrats, and 1 Independent?
Answer: Using the results from (a) and (b), the requested probability is 14,850/80,730 = 0.183946, or about 18.39%.
Problem: Write the first four numbers that appear in the row of Pascal's triangle that contains twenty numbers.
Solution (with communication): These numbers are
_{19}C_{0}, _{19}C_{1} , _{19}C_{2} , and _{19}C_{3 }.
They are 1, 19, 171, and 969. 
Problem: If four cards are randomly dealt from a regular deck of 52 cards, what is the probability that the hand contains one heart, one diamond, one spade, and one club?
Solution (with communication): The requested probability is
(_{13}C_{1})(_{13}C_{1})(_{13}C_{1}) (_{13}C_{1})/ (_{52}C_{4})
= 13^{4}/(270,725) = 0.105498, or about 10.55%. 
