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bwmwilson
Dustin has a set of 180 distinct blocks. Each of these blocks is made of either wood or plastic and comes in one of three sizes (small, medium, large), five colors (red, white, blue, yellow, green), and six shapes (triangular, square, rectangular, hexagonal, octagonal, circular). How many of the blocks in this set differ from a) the small red wooden square block in exactly one way? (For example, the small red plastic square block is one such block.) b) the large blue plastic hexagonal block in exactly two ways? (For example, the small red plastic hexagonal block is one such block.) listing the differences for part a) was simple but part b) I am unsure of
This question is interesting. So there are 4 sets of characteristics: Material, Size, Color, Shape. a) Small Red Wooden Square not Small = Medium & Large., so 2/3 of the blocks; OR not Red = the other 4 colors, so 4/5 of the blocks; OR not Wooden = Plastic, so 1/2 of the blocks; OR not Square = the other 5 shapes, so 5/6 of the blocks. This is where I'm stuck. So we pick the largest fraction now? 5/6? (5/6) * 180 = ??
no, you would add all them up, not as fractions. there are 2 differences in size, 4 in color, 1 in material, and 5 in shapes. Thats 12 varatations and what I did was just list them out. changing the shape: small, red, wood, triangular small, red, wood, rectangular small, red, wood, hexagonal small, red, wood, octagonal small, red, wood, circular changing the material: small, red, plastic, square changing the color: small, white, wood, square small, blue, wood, square small, yellow, wood, square small, green, wood, square changing the size: medium, red, wood, square large, red, wood, square This would not be as simple for part B though and I am unsure of where to start with that one.
b) Large Blue Plastic Hexagonal And different in exactly two ways so the other blocks could be: -Large Blue but not Plastic and not Hex -Large Plastic but not not Blue and not Hex -Large Hex but not Blue and not Plastic -Blue Plastic but not Large and not Hex -Blue Hex but not Large and not Plastic -Plastic Hex but not Large and not Blue Is that all? Did I miss a case? There must be a better way to do this...
OK so how many blocks are there in each variation for a)?
There are 12 variations with only one thing not in common from the original block for A
Using your list i took the number of possible differences for each and multiplied them. Once done for every catagory just used the rule of sum and I believe thats the answer but not positive. Large Blue but not Plastic and not Hex: 1*5 = 5 Large Plastic but not Blue and not Hex: 4*5 = 20 Large Hex but not Blue and not Plastic: 4*1 = 4 Blue Plastic but not Large and not Hex: 2*5 = 10 Blue Hex but not Large and not Plastic: 2*1 = 2 Plastic Hex but not Large and not Blue: 2*4 = 8 Total of 49 blocks
OK, two Materials, three Sizes, five Colors and six Shapes. 2*3*5*6 = 180 So isn't there only 1 block per variation because there are 180 blocks?
Not to sure where you are going with that. It it isnt hex then there are 5 different options it could be instead and so on for the other characteristics.
3 sizes 5 colors 6 shapes 2 materials s r t w m w s p l b r y h g o c 3*5*6*2 = 10*18 = 180 different elements ......................................... -l, -b, p, h -l, b, -p, h -l, b, p, -h l, -b, -p, h l, -b, p, -h l, b, -p, -h are all the sets that differ l,b,p,h by exactly 2 properties
the letters that are "positive" in my write up are the common properties and equate to 1 (since they are the only property that spot can have); the others reduce by one since they have to differ by 1. 3*5*6*2 ......................................... 2*4*1*1: 8 +2*1*5*1: 10 +2*1*1*1: 2 +1*4*5*1: 20 +1*4*1*1: 4 +1*1*5*1: 5 ------------ 39