#$&* course mth 152 9/9/10 around 7:00Pm Your solution, attempt at solution.
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Given Solution:: ** We can list the possibilities, or we can analyze the numbers. First we list: Using letters for the names, there are 12 possibilities (note that the secretary must be either k or e, the others chosen from a, b, d. The secretary, president and treasurer are listed in said order): eab, ead, eba, ebd, eda, edb, kab, kad, kba, kbd, kda, kdb. Next we analyze the numbers: There are two women, so two possibilities for the first person selected. The other two will be selected from among the three men, so there are 3 possibilities for the second person chosen, leaving 2 possibilities for the third. The number of possibilities is therefore 2 * 3 * 2 = 12. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question 11.1.12 and 18. In how many ways can the total of two dice equal 5? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 4 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: ** Listing possibilities on first then second die you can get 1,4, or 2,3 or 3,2 or 4,1. There are Four ways. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* ********************************************* question: In how many ways can the total of two dice equal 11? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 2 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: ** STUDENT SOLUTION AND INSTRUCTOR RESPONSE: There is only 1 way the two dice can equal 11 and that is if one lands on 5 and the other on 6 INSTRUCTOR RESPONSE: There's a first die and a second. You could imagine that they are painted different colors to distinguish them. You can get 5 on the first and 6 on the second, or vice versa. So there are two ways. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question 11.1.36 5-pointed star, number of complete triangles How many complete triangles are there in the star and how did you arrive at this number? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 5 after drawing a star you see that each point makes a triangle and the pentagon in the center also has 5 triangles inside of it. In total you have 10 triangles. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: ** If you look at the figure you see that it forms a pentagon in the middle (if you are standing at the very center you would be within this pentagon). Each side of the pentagon is the side of a unique triangle; the five triangles formed in this way are the 'spikes' of the star. Each side of the pentagon is also part of a longer segment running from one point of the start to another. This longer segment is part of a larger triangle whose vertices are the two points of the star and the vertex of the pentagon which lies opposite this side of the pentagon. There are no other triangles, so we have 5 + 5 = 10 triangles. STUDENT COMMENT I am sorry but I cannot see but 8 triangles the 5 spikes , but only 3 larger triangles that incorporate two spikes and the vertex just as you discussed. What am I missing????? INSTRUCTOR RESPONSE: I know your work well, and you are seldom wrong, so I went back to take another look, just to be sure. The figure fools the eye, and it fooled mine enough that I was just about convinced. I clearly saw just three larger triangles and couldn't make myself see five. So I started out my response believing that I had a long-standing error in the solution to this problem. And it took me awhile to convince myself that I was really right in the first place. Here's the reasoning that led me back to my original solution. My eye still doesn't really want to believe it, but I can draw the picture, and if I look at one vertex of the pentagon at a time, I can see it. I'm more convinced than ever that we can't believe our eyes. Here goes. You might want to draw the picture in order to follow the labels: If the points of the 'star' are A, B, C, D and E, in order as we go around the 'star', then each of these points is connected to exactly two of the others, and no point is connected to either of its 'nearest neighbors'. So A is connected to C and D B is connected to D and E C is connected to A and E D is connected to A and B E is connected to B and C. This gives us 10 line segments, namely AC, AD, BD, BE, CE, CA, DA, DB, EB and ED, each potentially the side of a triangle. However these 10 segments are redundant, as follows: AC AD BD BE CA (same segment as AC) CE DA (same segment as AD) DB (same segment as BD) EB (same segment as BE) EC (same segment as CE). Thus we have only five segments connecting the points of the star. Each of these segments forms the longest side of an iscosceles triangle, two of whose vertices are the two points of the 'star', and the third of which is a vertex of the pentagon. The pentagon has five vertices. If you look at each vertex in turn, you will see that it is the vertex of a triangle. So in addition to the five 'points' on the star, there are five triangles, one for each vertex. This raises the total number of triangles to 10. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question 11.1.40 4 x 4 grid of squares, how many squares in the figure? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 16 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: ** I count 16 small 1 x 1 squares, then 9 larger 2 x 2 squares (each would be made up of four of the small squares), 4 even larger 3 x 3 squares (each made up of nin small squares) and one 4 x 4 square (comprising the whole grid), for a total of 30 squares. Do you agree? ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question 11.1.50 In how many ways can 30 be written as sum of two primes? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: 7 + 23, 13 + 17, 11 + 19 confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: **STUDENT SOLTION AND INSTRUCTOR COMMENT: There are 4 ways 30 can be written as the sum of two prime numbers: • 30 = 29 + 1 (instructor note: this is not a sum of two primes) • 30 = 19 + 11 • 30 = 23 + 7 • 30 = 17 + 13 INSTRUCTOR COMMENT: Good, but 1 isn't a prime number. It only has one divisor. The rest of your answers are correct. All sums give you 30, and 7, 11, 13, 17, 19 and 23 are all prime numbers.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question 11.1.60 four adjacent switches; how many settings if no two adj can be off and no two adj can be on YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your Solution: There are only two ways to have the switches so that no two adjacent switches are in the same position. confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:: ** There are a total of 16 settings but only two have the given property of alternating off and on. If the first switch is off then the second is on so the third is off so the fourth is on. If the first is off then the second is on and the third is off so the fourth is on. So the two possibilities are off-on-off-on and on-off-on-off. If we use 0 to represent ‘off’ and 1 to represent ‘on’ these possibilities they are written 0101 and 1010. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* question Add comments on any surprises or insights you experienced as a result of this assignment. ** STUDENT COMMENT: No surprises and it's early so i'm reaching for insight as a child reaches for a warm bottle of milk Your comments or questions: