Assignment 74

 "Like the crest of a peacock so is mathematics at the head of all knowledge." - (Anonymous)   Ancient Egyptian Mathematics: Papyrus is a form of paper made from a reed that grows along the banks of the Nile River. Four very ancient documents provide us information about the amazing mathematics of the Egyptians. (Two will be mentioned here; the other two will be referenced in Assignment #75.) The Moscow Papyrus (ca 1850 B.C.): It is 18 feet long, 3 inches wide, and contains 25 problems. It is now in the Museum of Fine Arts in Moscow. The Rhind Papyrus (ca 1605 B.C.): Sometimes called the Ahmes Papyrus, it is 18 feet long and 13 inches wide. It is the most informative Egyptian document we have. It contains 85 problems. It resides in the British Museum. Research on these documents would represent time well spent for those who appreciate the value of mathematics. Why did Herkimer have trouble opening a can of soda pop during the second game of a baseball doubleheader? Answer: Because the home team lost the opener. Herky wants to know: If mail you sent out is damp when it is returned, is this because of postage dew? Why is it that a guy who says he will stop acting like a fool isn't acting? ASSIGNMENT #74 Reading: Section 12.1, pages 701-704. Written: Page 706/39-53(odds), 55-57, 62,63. You can write answers neatly in text.
 Mathematical word analysis:HELIX : This word helix comes from Greek origins. We think of helix as referencing something with a spiral form. The Greek word referenced concepts related to wrapping or twisting.
It is important to understand use of factorial (!), nCr, and nPr.

The number of ways three people could arrange themselves in a line is 3x2x1 = 3! = 6.

The number of ways fifteen people could arrange themselves in a line is 15x14x13x...x2x1 = 15!, which is approximately 1,307,674,368,000, or about 1.3x1012. If these people could demonstrate one arrangement per second, it would take them 41,466 years, or over 414 centuries.

The number of ways one can arrange the letters in ABCDEFGHIJK is 11! = 39,916,800.

The number of ways one can arrange the letters in MISSISSIPPI is 11!/(4!x4!x2!) = 34,650.

The number of ways you can arrange 3 books from a set of 7 on a bookshelf is 7P3 = 7x6x5 = 210.

The number of combinations of 3 books taken from a set of 7 books is 7C3 = 35.

 Problem: There are ten students in a math class. They plan to demonstrate all possible ways they can arrange themselves in a line. Assuming that they could display one arrangement per minute, how long would it take them to show all possible arrangements? Solution (with communication): The number of possible arrangements (and hence the number of minutes required) would be 10! = 3,628,800. This is equivalent to 60,480 hours, or 2,520 days, or 6.9 years. Problem: How many four digit numbers do not contain the digit 7? Solution (with communication): The left-most digit cannot be 0 or 7, and each of the remaining three digits cannot be 7. Hence, the number of four digit numbers that do not contain 7 is 8x9x9x9 = 5,832.