Color coding to get an FPT algoirthm for $k$ disjoint triangles

Consider the following problem:

Input: A graph $ G=(V,E)$ and an integer $ k \in \mathbb{N}$

Output: Are there $ k$ vertex-disjoint triangles in $ G$ ?

Assume we want to use color coding to develop an FPT Algorithm for that, as done here (starting from slide 60). The reference material proposes the following method:

  1. Choose a random coloring $ V \rightarrow [3k]$
  2. Check if there is a colorful solution where the $ 3k$ vertices of the $ k$ triangles use distinct colors.

For 2. it proposes this, amongst others, this method:

Try every permutation $ \pi$ of $ [3k]$ and check if there are triangles with colors $ (\pi(1), \pi(2), \pi(3)), (\pi(4),\pi(5),\pi(6), \dots)$

I don’t understand why we have to check every permutation $ \pi$ of the colors. Wouldn’t it be enough to just to check each triple of vertices, see if there are a triangle and if so, only count this triangle if it only uses colors we have not seen before? So like so:

  1. For each triple $ x,y,z \in V$ :
  2. If $ x,y,z$ form a triangle and colors $ {c(x),c(y),c(z)}$ not in colors_seen_so_far:

    2.1 colors_seen_so_far += $ \{c(x), c(y), c(z)\}$

    2.2 num_triangles += 1

where we initialize colors_seen_so_far = $ \emptyset$ and num_triangles = $ 0$