Number of ways to draw 3 balls from a box with at least one green ball when the box has 2 blue, 3 green and 4 yellow balls

Introduction / Context:
This question involves combinations with a restriction, which is a very common type in aptitude exams. You must count the number of ways to choose 3 balls from a set of coloured balls, under the condition that at least one of the chosen balls is green.

Given Data / Assumptions:

• The box contains 2 blue balls.
• The box contains 3 green balls.
• The box contains 4 yellow balls.
• All balls of the same colour are identical in colour but considered distinct objects for counting combinations.
• We must select exactly 3 balls.
• The selection must include at least one green ball.

Concept / Approach:
A standard strategy for at least one type problems is to first count all possible selections without any restriction and then subtract the selections that violate the condition. Here, the undesired selections are those that contain no green balls. We use the combination formula nCr to count selections.

Step-by-Step Solution:
Step 1: Total number of balls = 2 blue + 3 green + 4 yellow = 9.Step 2: Total ways to choose any 3 balls from 9 balls = 9C3.Step 3: Compute 9C3 = 9 * 8 * 7 / (3 * 2 * 1) = 84.Step 4: Now count the selections with no green ball at all.Step 5: Balls that are not green are 2 blue + 4 yellow = 6 balls.Step 6: Number of ways to choose 3 balls all from these 6 non green balls = 6C3 = 6 * 5 * 4 / 6 = 20.Step 7: Valid selections with at least one green ball = total selections minus selections with no green = 84 - 20 = 64.

Verification / Alternative check:
You could also directly split into cases: exactly 1 green, exactly 2 green, and exactly 3 green, and compute each using combinations. Adding those case counts should again give 64. If you do this and the sum matches 64, it confirms that the complement method is correct.

Why Other Options Are Wrong:

• 48 and 32 are smaller numbers that might arise from partial case counting but not from a complete and correct calculation.
• 24 is far too small and ignores many valid combinations.

Common Pitfalls:
Many students mistakenly treat colours as identical groups and do not count distinct balls separately, which leads to undercounting. Another typical error is to count only one or two cases, such as exactly one green ball, without including other valid cases. The complement method is usually easier and less error prone for at least one type questions.

Final Answer:
The number of ways to draw 3 balls including at least one green ball is 64, so the correct answer is 64.