A bag contains 2 red roses, 4 yellow roses and 6 pink roses. Two roses are drawn at random without replacement. What is the probability that the two roses drawn are not of the same colour?

Introduction / Context:
This question is about drawing two items without replacement from a collection with three different colours. Instead of asking for matching colours, it asks for the probability that the two drawn roses are not of the same colour. This can be solved directly or via the complement approach using probability of same colour first.

Given Data / Assumptions:

• Red roses = 2.
• Yellow roses = 4.
• Pink roses = 6.
• Total roses = 2 + 4 + 6 = 12.
• Two roses are drawn at random without replacement.
• Event of interest: the two roses are not of the same colour.

Concept / Approach:
It is usually easier to first find the probability that both roses are of the same colour, then subtract this from 1. Same colour means both red, both yellow or both pink. We compute this using combinations, then use complement: P(not same colour) = 1 minus P(same colour).

Step-by-Step Solution:
Total number of ways to choose 2 roses from 12 = C(12, 2) = 12 * 11 / 2 = 66. Ways to choose 2 red roses = C(2, 2) = 1. Ways to choose 2 yellow roses = C(4, 2) = 4 * 3 / 2 = 6. Ways to choose 2 pink roses = C(6, 2) = 6 * 5 / 2 = 15. Total same colour pairs = 1 + 6 + 15 = 22. P(same colour) = 22 / 66 = 1 / 3. P(not same colour) = 1 - 1 / 3 = 2 / 3.

Verification / Alternative check:
We could also compute P(not same colour) directly by summing probabilities of red yellow, red pink and yellow pink mixed pairs, using ordered draws and then converting to unordered results. The calculations are more tedious but lead to the same final fraction 2/3, confirming the complement approach is correct.

Why Other Options Are Wrong:
1/6 and 14/33 do not match either P(same colour) or P(not same colour) and come from incorrect counting. 5/6 is much too large; it would mean same colour is very unlikely, which is not supported by the counts.

Common Pitfalls:
Some students compute the probability of same colour and forget to subtract from 1. Others miscalculate combinations, especially for the pink roses. Carefully writing C(n, 2) and computing it helps prevent numerical errors.

Final Answer:
So, the probability that the two roses drawn are of different colours is 2/3.