Jolene makes a figure that is 5 cubes long, 5 cubes high, and 5 cubes wide. She paints the outside of the figure yellow. How many cubes have no yellow faces?

Question

A camp has 8 cabins. Each cabin has a direct path to each of the other cabins. How many paths are in the camp

Answer 1

draw a diagram

the 25 in the front are painted

the 25 in the back are painted

15 on the right side are painted (the 5 on each end, front and back, have been accounted for, for a total of 10 already accounted for)

15 on the left side are painted (same reasoning)

this takes care of all the cubes around the perimeter

so far 80 are painted

this leaves 45 inside the perimeter

the top 9 and the bottom 9 are painted

98 in all are painted yellow

there are 5*5*5=125 cubes, 98 of which are colored yellow

this leaves 27 cubes not colored yellow

 

cabin 8 has a path to 7 other cabins

cabin 7 has a path to 6 other cabins (not 7 because there is a path between cabin 8 and cabin 7 already)

cabin 6 has a path to 5 other cabins (same reasoning as above)

5 has a path to 4

4 has a path to 3

3 has a path to 2

2 has a path to 1

1 already has a path coming to it from all the other cabins

7+6+5+4+3+2+1=28 paths altogether

you can use the combination formula but way this is easier to understand