A box contains 5 green, 4 yellow and 3 white marbles. If three marbles are drawn at random, what is the probability that the three marbles are not all of the same colour?

Introduction / Context:
This question tests basic probability with combinations. Rather than directly counting the favourable cases, it is often easier to find the complement event (in this case, all marbles of the same colour) and subtract from 1. This technique simplifies calculations and is widely used in exam problems.

Given Data / Assumptions:

• Green marbles: 5.
• Yellow marbles: 4.
• White marbles: 3.
• Total marbles: 5 + 4 + 3 = 12.
• We draw 3 marbles at random without replacement.
• We want the probability that the 3 drawn marbles are not all of the same colour.

Concept / Approach:
Probability = (number of favourable outcomes) / (total number of possible outcomes). The total number of ways to choose 3 marbles out of 12 is 12C3. The unfavourable case is when all 3 marbles are of the same colour (either all green, all yellow or all white). We first compute the probability of this unfavourable event and subtract it from 1 to get the required probability that the three are not all of the same colour.

Step-by-Step Solution:
Total ways to draw 3 marbles: 12C3 = 12 * 11 * 10 / 6 = 220.Ways to draw 3 green marbles: 5C3 = 10.Ways to draw 3 yellow marbles: 4C3 = 4.Ways to draw 3 white marbles: 3C3 = 1.Total ways with all same colour = 10 + 4 + 1 = 15.Probability(all same colour) = 15 / 220.Required probability = 1 - 15 / 220 = (220 - 15) / 220 = 205 / 220.Simplify the fraction by dividing numerator and denominator by 5: 205 / 220 = 41 / 44.

Verification / Alternative check:
You could also directly count all non same colour combinations, but that requires many cases.The complement method is shorter and less error prone, and the final simplified fraction 41 / 44 lies between 0 and 1 as expected.

Why Other Options Are Wrong:
40/44 is slightly smaller and would correspond to 200 favourable outcomes instead of 205.44/41 is greater than 1, which is impossible for a probability.40/39 is also greater than 1, so it cannot represent a probability.

Common Pitfalls:
Forgetting that there are three colours and only counting one all same colour case.Dividing by the wrong total number of outcomes or miscomputing 12C3.Not simplifying the final fraction when the question options are given in lowest terms.

Final Answer:
The probability that the three marbles are not all of the same colour is 41/44.