(1) How many ways can you paint the faces of a cube with 2 colors? I ()()()()()() 1*64 = 64 3 Face 2 ()()( , , , ) 6* 8 = 48 1 ()()( , )( , ) 3*16 = 48 4 Corner 2 ( , , )( , ,) 8* 4 = 32 6 Edge ( , )( , )( , ) 6* 8 = 48 ----- ----- 24 240/24 = 10 <----answer (6,0) (5,1) (4,2) (3,3) 2 2 4 2 <---number of each type (2) How many ways can you paint the faces of a cube with 3 colors? I ()()()()()() 1*729 = 729 3 Face 2 ()()( , , , ) 6* 27 = 162 1 ()()( , )( , ) 3* 81 = 243 4 Corner 2 ( , , )( , ,) 8* 9 = 72 6 Edge ( , )( , )( , ) 6* 27 = 162 ----- ----- 24 1368/24 = 57 (3) How many 8 bead necklaces are there with 2 colors? I (12345678) 4 Edge Flips ( , )( , )( , )( , ) (1357)(2468) 4 Corner Flips ()()( , )( , )( , ) (14725836) (15)(26)(37)(48) (16385274) (1753)(2864) (18765432) 1 ()()()()()()()() 1*256 = 256 4 ( , , , , , , , ) 4* 2 = 8 2 ( , , , )( , , , ) 2* 4 = 8 5 ( , )( , )( , )( , ) 5* 16 = 80 4 ()()( , )( , )( , ) 4* 32 = 128 -------------- ---------- 16 480/16 = 30 <--answer (8,0) (7,1) (6,2) (5,3) (4,4) 2 2 8 10 8 (4) How many 8 bead necklaces are there with 3 colors? 1 ()()()()()()()() 1*6561 = 6561 4 ( , , , , , , , ) 4* 3 = 12 2 ( , , , )( , , , ) 2* 9 = 18 5 ( , )( , )( , )( , ) 5* 81 = 405 4 ()()( , )( , )( , ) 4* 243 = 972 -------------- ---------- answer 16 7968/16 = 498