A box contains 3 blue marbles, 4 red marbles, 6 green marbles and 2 yellow marbles. If two marbles are picked at random without replacement, what is the probability that both marbles picked are from the set of blue or yellow marbles (that is, each selected marble is either blue or yellow)?

Introduction / Context:
This problem checks basic probability with combinations, where we draw two objects from a group containing several categories. The goal is to find the probability that both drawn marbles come only from a specified subset of colours, namely blue and yellow. Such questions train students to correctly count favourable outcomes and total outcomes using combinations instead of confusing them with ordered selections.

Given Data / Assumptions:

• Blue marbles: 3.
• Red marbles: 4.
• Green marbles: 6.
• Yellow marbles: 2.
• Two marbles are drawn at random without replacement.
• We want both marbles to be either blue or yellow, with any combination among them allowed.

Concept / Approach:
The standard approach is to use combinations to count selections where order does not matter. The total number of ways to choose 2 marbles from all available marbles is computed using nC2. Then we count the favourable ways where both marbles are drawn only from the combined group of blue and yellow marbles. The probability is favourable combinations divided by total combinations.

Step-by-Step Solution:
Step 1: Count the total number of marbles: 3 + 4 + 6 + 2 = 15 marbles.Step 2: Total ways to choose 2 marbles from 15 marbles is 15C2 = (15 × 14) / 2 = 105.Step 3: The favourable colours are blue and yellow. Number of blue or yellow marbles combined is 3 + 2 = 5.Step 4: Number of ways to choose 2 marbles both from this group of 5 is 5C2 = (5 × 4) / 2 = 10.Step 5: Probability that both marbles are either blue or yellow is favourable / total = 10 / 105.Step 6: Simplify 10 / 105 by dividing numerator and denominator by 5, giving 2 / 21.
Verification / Alternative check:
We can also check the result by considering the probability step by step. First pick any marble that is blue or yellow: probability = 5 / 15. Then, given that, there are 4 remaining blue or yellow marbles out of 14 total marbles, so the second probability is 4 / 14. Multiply: (5 / 15) * (4 / 14) = (5 * 4) / (15 * 14) = 20 / 210 = 2 / 21. This matches the earlier result, confirming correctness.

Why Other Options Are Wrong:
3/17, 4/21 and 5/17 do not match the correct calculation and arise from incorrect counting or incorrect reduction of fractions. For example, 4/21 might come from mistaken counting of favourable cases. None of these values matches the properly derived probability 2/21. Therefore, option 2/21 is the only correct choice.

Common Pitfalls:
Students often mistakenly use permutations (which consider order) instead of combinations, or they fail to combine the blue and yellow marbles before counting. Another common error is to add probabilities instead of counting combinations. Always decide whether the order of selection matters and use combinations when order is irrelevant in selection problems like this one.

Final Answer:
The required probability that both marbles are either blue or yellow is 2/21.