Solution:

Total number of faces in a pentagonal prism = 2 pentagons + 5 rectangles = 7 faces
Therefore, we need to paint 7 faces with 7 different colours.
This is a permutation problem because the order in which we paint the faces matters.

Number of ways to paint the first face = 7
Number of ways to paint the second face = 6 (because one colour has already been used)
Number of ways to paint the third face = 5 (because two colours have already been used)
Number of ways to paint the fourth face = 4
Number of ways to paint the fifth face = 3
Number of ways to paint the sixth face = 2
Number of ways to paint the seventh face = 1

Total number of ways = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

Therefore, there are 5040 ways to paint the seven faces of a pentagonal prism with 7 different colours.

Explanation:

- We first break down the problem into smaller parts i.e. the number of ways to paint each face.
- We then use the multiplication principle of counting to find the total number of ways to paint all 7 faces.
- The multiplication principle of counting states that if there are m ways to do one thing and n ways to do another thing, then there are m x n ways to do both things.
- In this case, we multiply the number of ways to paint each face to get the total number of ways to paint all 7 faces.
- We can also verify our answer by using the formula for permutations with repetition, which is n^r, where n is the number of options and r is the number of choices.
- In this case, n = 7 and r = 7, so the formula gives us 7^7 = 5040, which matches our answer.