16. (2 marks) Each face and each vertex of a regular tetrahedron is coloured red or blue. How many different ways of colouring are there? (Two tetrahedrons are said to have the same colouring if we can rotate them suitably so that corresponding faces and vertices are of the same colour.
(2 分) 在正四面體的每一個頂點和每一個面都塗上紅色或藍色, 問有多少種不同的塗色方法? (若能將兩個已塗色的四面體適冨旋轉使得對應的頂點和面同色, 則這兩個四面體只能算為一種塗色方法 0 )
16. Let $x$ be the number of blue vertices and $y$ be the number of blue faces in a coloring. If $x=0$, we have 5 different ways to color the faces, namely $y=0,1,2,3,4$. If $x=1$, there are 2 ways to color the face opposite to the blue vertex and the remaining three faces may have $0,1,2$ or 3 blue faces. Thus there are $2 \times 4=8$ different ways in this case. If $x=2$, call the edge connecting the two blue vertices the " main edge " and the two faces common to this edge the " main faces". There are 3 ways to color the main faces, namely, both red, both blue, and one red and one blue. In the former two cases there are 3 ways to color the remaining faces, while in the last case there are four because reversing the color of the two faces does matter. Hence there are 10 ways in this case. By symmetry, there are 8 ways corresponding to the case $x=3$ and 5 ways corresponding to $x=4$. Hence the answer is $5+8+10+8+5=\underline{\underline{36}}$.