For a sports meet, a winners’ stand comprising three wooden blocks is in the following form:

There are six different colours available to choose from and each of the three wooden blocks is to be painted such that no two of them has the same colour. In how many different ways can the winners’ stand be painted?

The correct option is 120.

Step-by-Step Explanation:

1. Analyze the Winners' Stand:
As shown in the image, the winners' stand consists of three adjacent wooden blocks of different heights. Because these three blocks are distinct in height and position (representing first, second, and third place), they are treated as three distinct objects. Let us refer to them as Block 1, Block 2, and Block 3.

2. Understand the Constraints:
- There are 6 different colours available to choose from.
- Each block must be painted with a single colour.
- No two blocks can share the same colour (all three colours must be distinct).

3. Calculate the Number of Ways (using the Multiplication Principle):
- First Block: We can choose any of the 6 available colours. This gives us 6 options.
- Second Block: Since it cannot be the same colour as the first block, we have 5 remaining colours to choose from. This gives us 5 options.
- Third Block: Since it cannot be the same colour as the first two blocks, we have 4 remaining colours to choose from. This gives us 4 options.

By the fundamental counting principle, the total number of ways to paint the blocks is the product of the number of choices for each block:

$$\text{Total Ways}=6\times 5\times 4$$

$$\text{Total Ways}=120$$

Alternatively, this can be solved using the permutation formula to find the number of ways to choose and arrange 3 distinct colours out of 6:

$${}_{6}P_3=\frac{6!}{(6-3)!}=\frac{6\times 5\times 4\times 3!}{3!}=6\times 5\times 4=120$$

Thus, there are exactly 120 different ways to paint the winners' stand.