Question

Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is the minimum number of jellybeans she must take out of her pocket to ensure that she has one of each color?

• A. 4
• B. 8
• C. 12
• D. 13
• E. 16

Hint:

For problems that ask to "guarantee" or "ensure" a result, always think about the worst possible sequence of events. Add up the counts of the groups you want to avoid picking from first, then add 1.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a classic "worst-case scenario" problem, often related to the Pigeonhole Principle. To "ensure" an outcome, we must consider what would happen if we were as unlucky as possible.
Step 2: Key Formula or Approach:
To guarantee a specific combination, assume you first draw all the items you do *not* want, or all of the most numerous items. The very next draw will then guarantee the desired outcome.
Step 3: Detailed Explanation:
We want to guarantee that we have at least one of each color: red, green, and blue.
Let's imagine the worst possible luck.
Worst Case Scenario:
First, you draw all the jellybeans of the most common color. There are 8 red ones. So, you pull out all 8 red jellybeans. At this point, you have only one color.
Number of jellybeans drawn: 8 (all red).
Next, you draw all the jellybeans of the next most common color. Both green and blue have 4. Let's say you draw all the green ones.
Number of jellybeans drawn: 8 red + 4 green = 12.
After drawing 12 jellybeans, you have all the red and all the green ones, but you still haven't drawn a single blue jellybean. The only jellybeans left in the pocket are the 4 blue ones.
Therefore, the very next jellybean you draw (the 13th one) is guaranteed to be blue.
This will complete your set of at least one of each color.
Step 4: Final Answer:
Grace must take out a minimum of 13 jellybeans to be absolutely certain she has one of each color. The correct option is (D).