Correct Option
The problem requires determining the number of distinct ways to paint three wooden blocks, representing 1st, 2nd, and 3rd place, using three different colours selected from a pool of six available colours. This is a permutation problem, as the order of colours on the distinct blocks matters.
The calculation can be performed in two steps:
- Selection of colours: The number of ways to choose 3 distinct colours from 6 available colours is given by the combination formula nCk: 6C3 = 6! / (3! × (6-3)!) = 6! / (3! × 3!) = (6 × 5 × 4) / (3 × 2 × 1) = 20.
- Arrangement of colours: Once 3 colours are selected, they can be arranged on the 3 distinct wooden blocks in 3! ways: 3! = 3 × 2 × 1 = 6.
Therefore, the total number of different ways to paint the winners' stand is the product of the selection and arrangement possibilities:
Total ways = 6C3 × 3! = 20 × 6 = 120.
Alternatively, this can be directly calculated using the permutation formula P(n, k):
P(6, 3) = 6 × 5 × 4 = 120.
Incorrect Options
- Option 2 (81): This value does not correspond to the correct application of permutation or combination principles for selecting and arranging 3 distinct items from 6. It might result from an incorrect calculation such as 34 or other misinterpretations of the problem's combinatorial nature.
- Option 3 (66): This value is not derived from the appropriate combinatorial method for this problem. It does not represent the number of ways to select and arrange 3 distinct colours from 6, as required by the problem statement.
- Option 4 (36): This value is incorrect. It could arise from an erroneous calculation like 6 × 6, or 6 × 3!, which incorrectly applies the principles of selection and arrangement for the given conditions. It does not accurately account for the selection of three distinct colours from six and their subsequent arrangement on three distinct blocks.
