13. An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of $\sqrt{3}$ units from the starting point. Find the number of possible routes the ant has taken.
(2 marks)
An ant crawls along the edges of a cube with side length 1 unit. Starting from one of the vertices, in each minute the ant travels from one vertex to an adjacent vertex. After crawling for 7 minutes, the ant is at a distance of $\sqrt{3}$ units from the starting point. Find the number of possible routes the ant has taken.
13. 546
13. Suppose the ant starts at $(0,0,0)$ and stops at $(1,1,1)$. In each step, the ant changes exactly one of the $x$-, $y$ - or $z$ - coordinate. The change is uniquely determined: either from 0 to 1 or from 1 to 0. Also, the number of times of changing each coordinate is odd. Hence the number of changes in the three coordinates may be 5-1-1 or 3-3-1 (up to permutations). It follows that the answer is $3 \times C_{5}^{7}+3 \times C_{3}^{7} \times C_{1}^{4}=546$.